+ i*^RIHt^RIHt^RIHt^RIHt^RIH t H t H t ^RI H t ^RIHt^RIHtHt^R IHt^R IHt^R IHt^R IHt^R IHt^RIHt^RIH t ^RI!H"t"H#t#H$t$H%t%H&t&H't'H(t(^RI)H*t*H+t+H,t,H-t-H.t.H/t/H0t0H1t1^RI2H3t3^RI4H5t5^RI6H7t7^RI8H9t9H:t:^RI;HRt?R@Rlt@RtARtBRtCRARltDRARltER tFRBR!ltGR"tHRBR#ltIR$tJRCR&ltKR'tLRBR(ltMR)tNR*tOR+tPRBR,ltQRAR-ltRRAR.ltSR/tTRAR0ltUR1tVR2tW]<'d=]X!]Y!]:]=]>]?]A]B]D]E]F]G]H]I]K]M]O]@]P]Q]R]S]N]T]U344wt=t>t?tAtBtDtEtFtGtHtItKtMtOt@tPtQtRtStNtTtU]D]=3]E]=3]9.tZ]K]D]=3]E]=3]=.3t[]P]G]=3]P]G]J]=3]9.t\]B]J3]9.t]]B]A]B]M]B]O]B]=3t^]B]A]I]B]A]K3]D]F]K]B3]\]Z]H][]B]H]H]]3]9.t_R33R4lt`RDR5ltaR6P4tc]d!]X!]e!]c]X!]Y!]f!4P]c44444th]R74ti]R84tj]R94tkR@R:ltlR;tmR<tnR=toR>tpR?tqR%#)E) defaultdict)Add)cacheit)Expr)Factors gcd_terms factor_terms) expand_mul)Mul)piI)Pow)S)ordered)Dummy)sympify bottom_up)binomial)coshsinhtanhcothsechcschHyperbolicFunction)cossintancotseccscsqrtTrigonometricFunction) perfect_power)factor)greedy)identitydebug) SYMPY_DEBUGcZVP4P4P4#)zwSimplification of rational polynomials, trying to simplify the expression, e.g. combine things like 3*x + 2*x, etc.... )normalr&expandrvs&q/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/simplify/fu.pyTR0r1 s" 99;    & & ((cRp\W4#)zReplace sec, csc with 1/cos, 1/sin Examples ======== >>> from sympy.simplify.fu import TR1, sec, csc >>> from sympy.abc import x >>> TR1(2*csc(x) + sec(x)) 1/cos(x) + 2/sin(x) c*\V\4'd4VP^,p\P\ V4, #\V\ 4'd4VP^,p\P\V4, #V#r) isinstancer!argsrOnerr"rr/as& r0fTR1..f5s_ b#   A55Q<  C  A55Q<  r2rr/r;s& r0TR1r>)s R r2cRp\W4#)aReplace tan and cot with sin/cos and cos/sin Examples ======== >>> from sympy.simplify.fu import TR2 >>> from sympy.abc import x >>> from sympy import tan, cot, sin, cos >>> TR2(tan(x)) sin(x)/cos(x) >>> TR2(cot(x)) cos(x)/sin(x) >>> TR2(tan(tan(x) - sin(x)/cos(x))) 0 c\V\4'd/VP^,p\V4\ V4, #\V\ 4'd/VP^,p\ V4\V4, #V#r5)r6rr7rrr r9s& r0r;TR2..fSs_ b#   Aq6#a&= C  Aq6#a&=  r2rr=s& r0TR2rBAs$ R r2c&aV3Rlp\W4#)ayConverts ratios involving sin and cos as follows:: sin(x)/cos(x) -> tan(x) sin(x)/(cos(x) + 1) -> tan(x/2) if half=True Examples ======== >>> from sympy.simplify.fu import TR2i >>> from sympy.abc import x, a >>> from sympy import sin, cos >>> TR2i(sin(x)/cos(x)) tan(x) Powers of the numerator and denominator are also recognized >>> TR2i(sin(x)**2/(cos(x) + 1)**2, half=True) tan(x/2)**2 The transformation does not take place unless assumptions allow (i.e. the base must be positive or the exponent must be an integer for both numerator and denominator) >>> TR2i(sin(x)**a/(cos(x) + 1)**a) sin(x)**a/(cos(x) + 1)**a c D <a VP'gV#VP4wrVP'gVP'dV#V 3Rlo VP4p\ VP 44Uu.uF,pS !W1V,4'dKW1P V43NK. ppV'gV#VP4p\ VP 44Uu.uF,pS !W2V,4'dKW2P V43NK. ppV'gV#V V 3RlpV!W4V!W%4.pVEFp\V\4'd\VP^,RR7pW9dVW(,W,8XdCVP\VP^,4W,,4R;W&W(&KS 'ds^V,p W9daW),W,8XdKVP\VP^,^, 4W,,4R;W&W)&EKEK EK EK\V\4'd\VP^,RR7pW9d[W(,W,8XdEVP\VP^,4W,),4R;W&W(&EKEKEKS 'gEKVP'gEKVP^,\PJgEK\VP^,\4'gEK\VP^,P^,RR7pW9gEKQW(,W,8XgEKgW(,P 'gVP"'gEKVP\VP^,^, 4W,),4R;W&W(&EK V'd\%YqP'4U U u.uFwrV 'gKW,NK up p ,!\%VP'4U U u.uFwrV 'gKW,NK up p !, pT\%VU U u.uF wrW,NK up p !\%VU U u.uF wrW,NK up p !, ,pV#uupiuupiuup p iuup p iuup p iuup p i)c<VP;'gVP;'dVP\\39;'gS;'d}VP ;'di\ VP4^8;'dI\;QJd&RVP4F 'gK R# R#!RVP44#)c3"TFap\;QJd0R\P!V44F 'gK RM% RM!!R\P!V444xKc R#5i)c3"TFCp\V\4;'g'VP;'dVP\JxKE R#5iN)r6ris_Powbase).0ais& r0 8TR2i..f..ok...s=,*B#2s+KKryy/K/KRWW^K*sA A A TFN)anyr make_args)rLr:s& r0rN.TR2i..f..ok..sU=5;C,--*,CCC,--*,,,5;sA+%A+1A+TF) is_integer is_positivefuncrris_Addlenr7rP)kehalfs&&r0okTR2i..f..oks..??3*$>>*=*=*=*=AFF q *=*==56VV=  @   @ =56VV==  @r2c<.pVFkpVP'gK\VP4^8gK3S'd \V4M \ V4pWC8wgKYVP W434Km V'd|\ V4Fwpwr4WWBV&K \V!P4pVF>pW,W#,,pS!W64'dW`V&K,VP W634K@ ?R#R#)N) rVrWr7r&r append enumerater as_powers_dict) dddonenewkrXknewivrZr[s && r0 factorize"TR2i..f..factorizesD888AFF a(,6!9,q/Dy QI.  $-dOLAy"G%4Dz002AtwA!xx ! aV,  r2FevaluateN)is_Mulas_numer_denomis_Atomralistkeyspopr6rrr7r_rrVrr8rSrTr items)r/nrbrXndonercrhtr:a1brYr[rZs& @r0r;TR2i..f{syyyI  " 999 I @    (,QVVXJ1baDk!UU1XJI   (,QVVXJ1baDk!UU1XJI & !! A!S!!q E26adadlHHS^QT12"&&AD14QBw15AD=#affQik"2QT!9:'++qu$1w As##q E26adadlHHS^adU23"&&AD14+6!(((qvvayAEE'9qvvay#..q q)E:6adadl HHS1-u45"&&AD14-0 qWWYqwwy6ytqAdaddy678B #//06Nqtt6N1OO OB AK Kn=6/6Ns<TT)TT T  T  T  T 7T Tr)r/rZr;s&f r0TR2iry_s8Sj R r2cZa^RIHoV3RlpVPRR4p\W4#)aInduced formula: example sin(-a) = -sin(a) Examples ======== >>> from sympy.simplify.fu import TR3 >>> from sympy.abc import x, y >>> from sympy import pi >>> from sympy import cos >>> TR3(cos(y - x*(y - x))) cos(x*(x - y) + y) >>> cos(pi/2 + x) -sin(x) >>> cos(30*pi/2 + x) -cos(x) signsimpc <\V\4'gV#VPS!VP^,44p\V\4'gV#VP^,\P ^, , P \P ^, VP^,, P u;JdRJdMV#\\\\\\\\\\\\/pV\V4,!\P ^, VP^,, 4pV#rT)r6r$rUr7rPirTrrrr r!r"type)r/fmapr|s& r0r;TR3..fs"344I WWXbggaj) *"344I GGAJa  , ,a"''!*1D0Q0Q YUY Y c3S#sCc3ODd2hQ 34B r2c"\V\4#rI)r6r$xs&r0TR3..s *Q 56r2c*VPRR4#)cBVP;'d VP#rI) is_numberrlrss&r0r'TR3....sakk..ahh.r2c6VP!VP!#rIrUr7rs&r0rrsaffaffor2replacers&r0rrs!)) . %'r2)sympy.simplify.simplifyr|rr)r/r;r|s& @r0TR3rs2$1  6 ' (B R r2c*VPRR4#)aTIdentify values of special angles. a= 0 pi/6 pi/4 pi/3 pi/2 ---------------------------------------------------- sin(a) 0 1/2 sqrt(2)/2 sqrt(3)/2 1 cos(a) 1 sqrt(3)/2 sqrt(2)/2 1/2 0 tan(a) 0 sqt(3)/3 1 sqrt(3) -- Examples ======== >>> from sympy import pi >>> from sympy import cos, sin, tan, cot >>> for s in (0, pi/6, pi/4, pi/3, pi/2): ... print('%s %s %s %s' % (cos(s), sin(s), tan(s), cot(s))) ... 1 0 0 zoo sqrt(3)/2 1/2 sqrt(3)/3 sqrt(3) sqrt(2)/2 sqrt(2)/2 1 1 1/2 sqrt(3)/2 sqrt(3) sqrt(3)/3 0 1 zoo 0 c\V\4;'dBVP^,\, ;pP;'dVP R9#)r)r^rF)r6r$r7r is_Rationalq)rrs& r0rTR4..sP q/ 0 E Eq " _Q ) ) E E./cc_.D Er2cVPVP^,P!VP^,P!4#r5rrs&r0rr s+ FF166!9>>166!9>>2 3r2rr.s&r0TR4rs 0 :: E 4  55r2c6aaaaaVVVVV3Rlp\W4#)aHelper for TR5 and TR6 to replace f**2 with h(g**2) Options ======= max : controls size of exponent that can appear on f e.g. if max=4 then f**4 will be changed to h(g**2)**2. pow : controls whether the exponent must be a perfect power of 2 e.g. if pow=True (and max >= 6) then f**6 will not be changed but f**8 will be changed to h(g**2)**4 >>> from sympy.simplify.fu import _TR56 as T >>> from sympy.abc import x >>> from sympy import sin, cos >>> h = lambda x: 1 - x >>> T(sin(x)**3, sin, cos, h, 4, False) (1 - cos(x)**2)*sin(x) >>> T(sin(x)**6, sin, cos, h, 6, False) (1 - cos(x)**2)**3 >>> T(sin(x)**6, sin, cos, h, 6, True) sin(x)**6 >>> T(sin(x)**8, sin, cos, h, 10, True) (1 - cos(x)**2)**4 c<VP'dVPPS8XgV#VPP'gV#VP^8R8XdV#VPS8R8XdV#VP^8XdV#VP^8Xd1S!S!VPP ^,4^,4#VP^,^8XdsVP^,pS!VPP ^,4S!S!VPP ^,4^,4V,,#VP^8Xd^pMiS'g0VP^,'dV#VP^,pM2\ VP4pV'gV#VP^,pS!S!VPP ^,4^,4V,#r~)rJrKrUexpis_realr7r%)r/rYpr;ghmaxpows& r0_f_TR56.._f>sm  bgglla/Ivv~~~I FFQJ4 I FFSLT !I 66Q;I 66Q;Qrww||A'*+ +vvzQFFAIa)!Abggll1o,>,A*BA*EEE166A::IFFAI!"&&)IFFAIQrww||A'*+Q. .r2r)r/r;rrrrrs&fffff r0_TR56r$s4!/!/F R r2c 4\V\\RWR7#)aMReplacement of sin**2 with 1 - cos(x)**2. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR5 >>> from sympy.abc import x >>> from sympy import sin >>> TR5(sin(x)**2) 1 - cos(x)**2 >>> TR5(sin(x)**-2) # unchanged sin(x)**(-2) >>> TR5(sin(x)**4) (1 - cos(x)**2)**2 c^V, #r^rs&r0rTR5..vQr2rr)rrrr/rrs&&&r0TR5rd$ S#C AAr2c 4\V\\RWR7#)aLReplacement of cos**2 with 1 - sin(x)**2. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR6 >>> from sympy.abc import x >>> from sympy import cos >>> TR6(cos(x)**2) 1 - sin(x)**2 >>> TR6(cos(x)**-2) #unchanged cos(x)**(-2) >>> TR6(cos(x)**4) (1 - sin(x)**2)**2 c^V, #rrrs&r0rTR6..rr2r)rrrrs&&&r0TR6ryrr2cRp\W4#)zLowering the degree of cos(x)**2. Examples ======== >>> from sympy.simplify.fu import TR7 >>> from sympy.abc import x >>> from sympy import cos >>> TR7(cos(x)**2) cos(2*x)/2 + 1/2 >>> TR7(cos(x)**2 + 1) cos(2*x)/2 + 3/2 cVP'd1VPP\8XdVP^8XgV#^\^VPP ^,,4,^, #rF)rJrKrUrrr7r.s&r0r;TR7..fsO bggllc1bffkIC"'',,q/)**A--r2rr=s& r0TR7rs . R r2c&aV3Rlp\W4#)aAConverting products of ``cos`` and/or ``sin`` to a sum or difference of ``cos`` and or ``sin`` terms. Examples ======== >>> from sympy.simplify.fu import TR8 >>> from sympy import cos, sin >>> TR8(cos(2)*cos(3)) cos(5)/2 + cos(1)/2 >>> TR8(cos(2)*sin(3)) sin(5)/2 + sin(1)/2 >>> TR8(sin(2)*sin(3)) -cos(5)/2 + cos(1)/2 c <VP'grVP'd^VPP\\ 39d9VP P'gVPP'gV#S 'dVP4Uu.uFp\V4NK upwr#\VRR7p\VRR7pWB8wgWS8wd\WE, 4pVP'dxVP^,P'dU\VP4^8Xd;VP^,P 'd\#VP%4!pV#\.\ .R./p\"P&!V4EF*pVP\\ 39d5V\)V4,P+VP^,4KSVP'dVP P,'dVP ^8dVPP\\ 39d\V\)VP4,P/VPP^,.VP ,4EKVR,P+V4EK- V\,pV\ ,p V'd V 'g#\V4^8g\V 4^8gV#VR,p\1\V4\V 44p\3V4F`pV P54p VP54p VP+\ W,4\ W, 4,^, 4Kb \V4^8d`VP54p VP54p VP+\ W,4\ W, 4,^, 4KoV'd)VP+\ VP5444\V 4^8daV P54p V P54p VP+\ W,4)\ W, 4,^, 4KpV 'd)VP+\ V P5444\\\#V!44#uupi)FfirstN)rlrJrKrUrrrrSrTrmr TR8rr7rrWrVr as_coeff_MulrQrr_ is_Integerextendminrangerq) r/rfrsrbnewnnewdr7r:csrva2rs & r0r;TR8..fs3 III III GGLLS#J & VV   "''"5"5"5I +-+<+<+>?+>aJqM+>?DAq&Dq&DyDIty)999!7!7!7BGG )bggaj.?.?.?boo/0BIRb$+r"Avv#s#T!W $$QVVAY/(((quu///AEEAIFFKKC:-T!&&\"))166;;q>*:155*@AT !!!$# I Ia3q6A:Q!IDz AA qABB KKRWBG 4a7 8!fqjBB KKRWBG 4a7 8 KKAEEG %!fqjBB KK#bg,RW5q8 9 KKAEEG %:c4j)**Y@sT rr/rr;s&f r0rrs"5+n R r2cRp\W4#)a'Sum of ``cos`` or ``sin`` terms as a product of ``cos`` or ``sin``. Examples ======== >>> from sympy.simplify.fu import TR9 >>> from sympy import cos, sin >>> TR9(cos(1) + cos(2)) 2*cos(1/2)*cos(3/2) >>> TR9(cos(1) + 2*sin(1) + 2*sin(2)) cos(1) + 4*sin(3/2)*cos(1/2) If no change is made by TR9, no re-arrangement of the expression will be made. For example, though factoring of common term is attempted, if the factored expression was not changed, the original expression will be returned: >>> TR9(cos(3) + cos(3)*cos(2)) cos(3) + cos(2)*cos(3) cTaVP'gV#RV3Rllo\VS4#)Tc<VP'gV#\\VP44p\ V4^8wdRp\ \ V44FepW$,pVfK\ V^,\ V44F4pW&,pVfKWW,pS!V4p W8wgK)WV&RW&&RpKc Kg V'd=\ VU u.uFq'gK V NK up !pVP'd S!V4pV#\V!p V 'gV#V wrrppV'dW8XdLW,^,\VV,^, 4,\VV, ^, 4,#V ^8dTTppRV ,\VV,^, 4,\VV, ^, 4,#W8XdLW,^,\VV,^, 4,\VV, ^, 4,#V ^8dTTpp^V ,\VV,^, 4,\VV, ^, 4,#uup i)rFFNT rVrorr7rWrr trig_splitrr)r/rr7hitrfrMjajwasnewrsplitgcdn1n2r:rwiscosdos&& r0rTR9..f..do s999 ()D4yA~s4y)ABz "1q5#d)4!W:$ g g:&)G&*DG"&C!5 *D7DbBrrD78ByyyV %E ', $CRAu86!8CQ N23Aqy>AA6aqA#vc1q5!)n,S!a%^;;86!8CQ N23Aqy>AA6aqAuS!a%^+CQ N::18s  I-"I-T)rVprocess_common_addends)r/rs&@r0r;TR9..fs&yyyI< ;|&b"--r2rr=s& r0TR9rs.B.H R r2c&aV3Rlp\W4#)awSeparate sums in ``cos`` and ``sin``. Examples ======== >>> from sympy.simplify.fu import TR10 >>> from sympy.abc import a, b, c >>> from sympy import cos, sin >>> TR10(cos(a + b)) -sin(a)*sin(b) + cos(a)*cos(b) >>> TR10(sin(a + b)) sin(a)*cos(b) + sin(b)*cos(a) >>> TR10(sin(a + b + c)) (-sin(a)*sin(b) + cos(a)*cos(b))*sin(c) + (sin(a)*cos(b) + sin(b)*cos(a))*cos(c) c<VP\\39dV#VPpVP^,pVP'EdS'd \ \ VP44pM\ VP4pVP4p\P!V4pVP'dV\8XdR\V4\\V4RR7,\V4\\V4RR7,,#\V4\\V4RR7,\V4\\V4RR7,, #V\8Xd<\V4\V4,\V4\V4,,#\V4\V4,\V4\V4,, #V#rFr) rUrrr7rVrorrqr _from_argsTR10)r/r;argr7r:rwrs& r0r;TR10..fasI 773* $I GGggaj :::GCHH-.CHH~ At$Axxx8q6$s1vU";;AtCF%8899q6$s1vU";;AtCF%88998q6#a&=3q6#a&=88q6#a&=3q6#a&=88 r2rrs&f r0rrOs$6 R r2cRp\W4#)a^Sum of products to function of sum. Examples ======== >>> from sympy.simplify.fu import TR10i >>> from sympy import cos, sin, sqrt >>> from sympy.abc import x >>> TR10i(cos(1)*cos(3) + sin(1)*sin(3)) cos(2) >>> TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3)) cos(3) + sin(4) >>> TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x) 2*sqrt(2)*x*sin(x + pi/6) c .a VP'gV#RV 3Rllo \VS R4pVP'EdL\\4pVPFp^pVP 'dVPFnpVP 'gKVP\PJgK7VPP'gKUW,PV4^pM V'dKV\P,PV4K .pVFp\4V,\43FpWa9gK \!\#W,44FpW,V,fK\!\#W,44FzpW,V,fK\%W,V,W,V,,4p S !V 4p W8wgKRVPV 4RW,V&RW,V&K K K K V'dT\%TVP'4U U u.uF%p \%V U u.uFq'gK V NK up !NK' up p ,!pEKSS !V4pV#V#uup iuup p i)Tcd<VP'gV#\\VP44p\ V4^8wdRp\ \ V44FepW$,pVfK\ V^,\ V44F4pW&,pVfKWW,pS!V4p W8wgK)WV&RW&&RpKc Kg V'd=\ VU u.uFq'gK V NK up !pVP'd S!V4pV#\VRR/p V 'gV#V wrrppV'dAW,p W8XdV \VV, 4,#V \VV,4,#W,p W8XdV \VV,4,#V \VV, 4,#uup i)rFFNTtwor)r/rr7rrfrMrrrrrrrrrr:rwsamers&& r0rTR10i..f..dosu999 ()D4yA~s4y)ABz "1q5#d)4!W:$ g g:&)G&*DG"&C!5 *D7DbBrrD78ByyyV /$/E &+ #CRAtf8s1q5z>)3q1u:~%f8s1q5z>)3q1u:~%-8s  F-"F-c>\\VP44#rI)tupler free_symbolsrs&r0r"TR10i..f..seGANN$;.fsyyyI6 &p$ <> iii%EWW888ff999166)9 " 2 2 2!I,,Q/"#C! % s!%%L''*D (1*ik2Az!&s58}!5A$x{2 (%*3ux=%9#(8A;#6$,&)%(1+ *C&D&(g#&:$(KK$426EHQK26EHQK$)&:"63 4"\\^#-+$'a(>a2a(>#?+#--/V r )?#-sJ  J !J ' J J rr=s& r0TR10irs$iV R r2Nc&aV3Rlp\W4#)aFunction of double angle to product. The ``base`` argument can be used to indicate what is the un-doubled argument, e.g. if 3*pi/7 is the base then cosine and sine functions with argument 6*pi/7 will be replaced. Examples ======== >>> from sympy.simplify.fu import TR11 >>> from sympy import cos, sin, pi >>> from sympy.abc import x >>> TR11(sin(2*x)) 2*sin(x)*cos(x) >>> TR11(cos(2*x)) -sin(x)**2 + cos(x)**2 >>> TR11(sin(4*x)) 4*(-sin(x)**2 + cos(x)**2)*sin(x)*cos(x) >>> TR11(sin(4*x/3)) 4*(-sin(x/3)**2 + cos(x/3)**2)*sin(x/3)*cos(x/3) If the arguments are simply integers, no change is made unless a base is provided: >>> TR11(cos(2)) cos(2) >>> TR11(cos(4), 2) -sin(2)**2 + cos(2)**2 There is a subtle issue here in that autosimplification will convert some higher angles to lower angles >>> cos(6*pi/7) + cos(3*pi/7) -cos(pi/7) + cos(3*pi/7) The 6*pi/7 angle is now pi/7 but can be targeted with TR11 by supplying the 3*pi/7 base: >>> TR11(_, 3*pi/7) -sin(3*pi/7)**2 + cos(3*pi/7)**2 + cos(3*pi/7) c <VP\\39dV#S'dVPpV!S^,4p\PpVP 'dVP 4wr2VP\\39dV#VP^,VP^,8XdV\S4p\S4pV\JdV^,V^,, V, #^V,V,V, #V#VP^,P'gVP^,P RR7wrFVP^,^8XdVP^,V,VP, p\\V44p\\V44pVP\8Xd^V,V,pV#V^,V^,, pV#)rFT)rational) rUrrrr8rlrr7 is_NumberrrTR11) r/r;rucorrmrrKs & r0r;TR11..f)sp 773* $I A$q& ABxxx(vvc3Z' wwqzQVVAY&II8qD1a4K++Q3q58OI%%%771:**D*9DAssQw!|cc1fQhqsslSNSN77c>1QB A1B r2r)r/rKr;s&f r0rrsT!F R r2cRp\W4#)a Helper for TR11 to find half-arguments for sin in factors of num/den that appear in cos or sin factors in the den/num. Examples ======== >>> from sympy.simplify.fu import TR11, _TR11 >>> from sympy import cos, sin >>> from sympy.abc import x >>> TR11(sin(x/3)/(cos(x/6))) sin(x/3)/cos(x/6) >>> _TR11(sin(x/3)/(cos(x/6))) 2*sin(x/6) >>> TR11(sin(x/6)/(sin(x/3))) sin(x/6)/sin(x/3) >>> _TR11(sin(x/6)/(sin(x/3))) 1/(2*cos(x/6)) c\V\4'gV#Rp\WP44wr#RpV!WV4pV!WV4pV#)cZ\\4p\P!V4FpVP 4wr4VP 'gK)V^8gK2VP \\39gKOV\V4,PVP^,4K V#r5) rsetr rQ as_base_exprrUrrraddr7)flatr7firwrYs& r0 sincos_args%_TR11..f..sincos_argshsss#DmmD)~~'<<\W4pW%,PV4Kb V#r)rrrremove)r/num_argsden_argsnargrZrUs&&& r0 handle_match&_TR11..f..handle_matchts^! AvC=(Dc]*D"^%%d+&Ir2)r6rmaprm)r/r rrrs& r0r;_TR11..fdsU"d##I !.?.?.AB  " 1 " 1 r2rr=s& r0_TR11rOs*#J R r2c&aV3Rlp\W4#)zSeparate sums in ``tan``. Examples ======== >>> from sympy.abc import x, y >>> from sympy import tan >>> from sympy.simplify.fu import TR12 >>> TR12(tan(x + y)) (tan(x) + tan(y))/(-tan(x)*tan(y) + 1) c <VP\8XgV#VP^,pVP'dS'd \ \ VP44pM\ VP4pVP 4p\P!V4pVP'd\\V4RR7pM \V4p\V4V,^\V4V,, , #V#r) rUrr7rVrorrqrrTR12)r/rr7r:rwtbrs& r0r;TR12..fsww#~Iggaj :::GCHH-.CHH~ At$Axxx#a&.VFRK!c!fRi-0 0 r2rrs&f r0rrs& R r2cRp\W4#)aCombine tan arguments as (tan(y) + tan(x))/(tan(x)*tan(y) - 1) -> -tan(x + y). Examples ======== >>> from sympy.simplify.fu import TR12i >>> from sympy import tan >>> from sympy.abc import a, b, c >>> ta, tb, tc = [tan(i) for i in (a, b, c)] >>> TR12i((ta + tb)/(-ta*tb + 1)) tan(a + b) >>> TR12i((ta + tb)/(ta*tb - 1)) -tan(a + b) >>> TR12i((-ta - tb)/(ta*tb - 1)) tan(a + b) >>> eq = (ta + tb)/(-ta*tb + 1)**2*(-3*ta - 3*tc)/(2*(ta*tc - 1)) >>> TR12i(eq.expand()) -3*tan(a + b)*tan(a + c)/(2*(tan(a) + tan(b) - 1)) c VP'g'VP'gVP'gV#VP4wrVP'dVP'gV#/pRp\ \ P!V44p\V4EFwrgV!V4pV'dPVwr\V PU u.uFqP^,NK up !p \PW<&WV&KeVP'dO\V4pVP'd0VPVP4\PWV&KKVP'gKVPP'g VP P"'gEKV!VP 4pV'd]Vwr\V PU u.uFqP^,NK up !p VPW<&WP,WV&EK\V4pVP'gEKVPVP4\PWV&EK V'gV#Rp\ \ P!\%V444p Rp\V 4EFwroV!V4pV'EgKV!V)4pV'd\P&W&EM7VP'dM\V4pVP'd.V PVP4\PW&KVP'dVPP'gVP P"'d{V!VP 4pV'd\PW&Mb\V4pVP'd.V PVP4\PW&EK^EKa\PW&Rp\VU u.uFqP^,NK up !p W<,pVP)\P4pVeV'dVW<&MVP+V 4W;;,\-V 4),uu&EK V'd\ V !\ V!, \ VP/4UUUu.uF@wpp\VPUu.uFp\-V4NK up!^, V,NKB uppp!, pV#uup iuup iuup iuupiuupppi)ch\V4pV'dVwr#pV\PJdVP'dp\ VP 4^8XdT\ ;QJd&RVP 4F 'dK RM RM!RVP 44'dW#3#R#R#R#R#R#)rFc3B"TFp\V\4xK R#5irI)r6r)rLr s& r0rN/TR12i..f..ok..sA&BJr3//&sFTN) as_f_sign_1r NegativeOnerlrWr7all)dirrr;rs& r0r[TR12i..f..oksBAa %!(((s166{a7GA!&&AA!&&AAA4KB8H(%r2cVP'd^\VP4^8XdBVPwr\V\4'd\V\4'dW3#R#R#R#R#)rFN)rVrWr7r6r)nir:rws& r0r[r&sTyyyS\Q.wwa%%*Q*<*<4K+=%/yr2FT)rVrlrJrmr7ror rQr`rrr8r&rrrSrKrTr r#extract_additivelyrqrrr)r/rsrbdokr[d_argsrfr%rrru_rn_argsrr(ednewedrYr:s& r0r;TR12i..fs RYYY")))I  "vvvQVVVI cmmA&'v&EA2AQVV4V&&))V45q yyyBZ999MM"''* !FI 1 1 1RWW5H5H5HrwwKDA8AffQii89AVVCF !66 FIByyy bgg.$%EE 1'2I cmmLO45v&EA2A1sG ! FIyyy#BZ999"MM"''2()FI FF---1D1D1DrwwK()FI!'B!yyy & bgg 6,-EE $ EE C+AffQii+,AB))!%%0E "CFGGAJ I#a& IK'N fc6l*3=@YY[1J=HTQ36 !8( &1A8(3)+,3-/02121=H1J,KKB U59^,8(1Js* U %U <U ?U(U#.U(#U(rr=s& r0TR12ir1s*aF R r2cRp\W4#)zChange products of ``tan`` or ``cot``. Examples ======== >>> from sympy.simplify.fu import TR13 >>> from sympy import tan, cot >>> TR13(tan(3)*tan(2)) -tan(2)/tan(5) - tan(3)/tan(5) + 1 >>> TR13(cot(3)*cot(2)) cot(2)*cot(5) + 1 + cot(3)*cot(5) c VP'gV#\.\.R./p\P!V4FjpVP \\39d5V\ V4,PVP^,4KRVR,PV4Kl V\,pV\,p\V4^8d\V4^8dV#VR,p\V4^8dVP4pVP4pVP^\V4\WV,4, \V4\WV,4, ,, 4KV'd)VP\VP444\V4^8dVP4pVP4pVP^\V4\WV,4,,\V4\WV,4,,4KV'd)VP\VP444\V!#rI) rlrr r rQrUrr_r7rWrq)r/r7r:rurt1t2s& r0r;TR13..f8syyyIRb$+r"Avv#s#T!W $$QVVAY/T !!!$ # I I q6A:#a&1*IDz!fqjBB KKSWS\1CGCL4HHI J KKAEEG %!fqjBB KKCGCL003r73rw<3GG H KKAEEG %Dzr2rr=s& r0TR13r7*s< R r2c,aRV3Rllo\VS4#)a Returns cos(x)*cos(2*x)*...*cos(2**(k-1)*x) -> sin(2**k*x)/(2**k*sin(x)) Examples ======== >>> from sympy.simplify.fu import TRmorrie, TR8, TR3 >>> from sympy.abc import x >>> from sympy import Mul, cos, pi >>> TRmorrie(cos(x)*cos(2*x)) sin(4*x)/(4*sin(x)) >>> TRmorrie(7*Mul(*[cos(x) for x in range(10)])) 7*sin(12)*sin(16)*cos(5)*cos(7)*cos(9)/(64*sin(1)*sin(3)) Sometimes autosimplification will cause a power to be not recognized. e.g. in the following, cos(4*pi/7) automatically simplifies to -cos(3*pi/7) so only 2 of the 3 terms are recognized: >>> TRmorrie(cos(pi/7)*cos(2*pi/7)*cos(4*pi/7)) -sin(3*pi/7)*cos(3*pi/7)/(4*sin(pi/7)) A touch by TR8 resolves the expression to a Rational >>> TR8(_) -1/8 In this case, if eq is unsimplified, the answer is obtained directly: >>> eq = cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9) >>> TRmorrie(eq) 1/16 But if angles are made canonical with TR3 then the answer is not simplified without further work: >>> TR3(eq) sin(pi/18)*cos(pi/9)*cos(2*pi/9)/2 >>> TRmorrie(_) sin(pi/18)*sin(4*pi/9)/(8*sin(pi/9)) >>> TR8(_) cos(7*pi/18)/(16*sin(pi/9)) >>> TR3(_) 1/16 The original expression would have resolve to 1/16 directly with TR8, however: >>> TR8(eq) 1/16 References ========== .. [1] https://en.wikipedia.org/wiki/Morrie%27s_law c <<VP'gV#V'd*VP4wr#S!V^4S!V^4, #\\4p/p.pVPFpVP 4wrV P 'dW\V\4'dAVP^,P4wrWK,PV 4WV&K}VPV4K .p VEFp WK,pVP4V'gK&^p V^,;rW9dV ^, p V^,pKV ^8Ed3\^V ,V,V ,4^V ,, \W,4, pRp.p\V 4FTpV^,p\W,RR7pVPV4\VV,T;'g VV,4pKV \V 4F]pVP4p\W,RR7pVV;;,V,uu&VV,'dKLVP!V4K_ V PVV,4EKj\VP^4V ,4pVPWV,,4EK V 'dI\#W,VU U u.uF%qV ,Fp \W,RR7NK K' up p ,!pV#uup p i)rNFrj)rlrmrror7rrr6rrr_sortrrrrqrr )r/rrsrbr7cossotherrrwrYrr:rrXcccinewargtakeccsrfkeyr;s&& r0r;TRmorrie..fscyyyI $$&DAQ71Q7? "4 A==?DA||| 1c 2 2q ..0r"Q QAA FFH!A$gFA!GBq5 Ab^AqD0RT:FDC"1Xa!!$7 2"49d.?.?d3i@ & #1X WWY!!$7S T) #CyyHHRL & JJvt|,AEE!HQJALLG,;> s{26&I26QQ1AC%(($&IIKB &Is+L rrr=s&@r0TRmorrierDYsv7r R r2c&aV3Rlp\W4#)aConvert factored powers of sin and cos identities into simpler expressions. Examples ======== >>> from sympy.simplify.fu import TR14 >>> from sympy.abc import x, y >>> from sympy import cos, sin >>> TR14((cos(x) - 1)*(cos(x) + 1)) -sin(x)**2 >>> TR14((sin(x) - 1)*(sin(x) + 1)) -cos(x)**2 >>> p1 = (cos(x) + 1)*(cos(x) - 1) >>> p2 = (cos(y) - 1)*2*(cos(y) + 1) >>> p3 = (3*(cos(y) - 1))*(3*(cos(y) + 1)) >>> TR14(p1*p2*p3*(x - 1)) -18*(x - 1)*sin(x)**2*sin(y)**4 c  <VP'gV#S'dWVP4wrV\PJd1\ VRR7p\ VRR7pW18wgWB8wd W4, pV#.p.pVP EF pVP 'dMVP4wrV P'g&VP'gVPV4K_TpM\Pp \V4p V 'd#V ^,P\\39d@V \PJdVPV4MVPWy,4KV wrp VPWPWW34EK \!\#V44p\%V4p\!\'^44;pwp prrV'EdVP)^4pV'EdV^,pVV ,P'EdVV ,P'Ed{VV ,VV ,8XEddVV ,VV ,8wEdNVP)^4p\+VV ,VV ,4pVV ,V8wdAVUu.uF pVV,NK ppVV ;;,V,uu&VP-^V4MMVV ,V8wd@VUu.uF pVV,NK ppVV ;;,V,uu&VP-^V4\/VV ,\4'd\pM\pVPVV ,)VV ,,V!VV ,P ^,4^,,V,4EKMVV ,VV ,8XdVV ,VV ,8XdVV ,VV ,8wdVP)^4pVV ,p\/VV ,\4'd\pM\pVPVV ,)VV ,,V!VV ,P ^,4^,,V,4EKVPVV,VV ,,4EK\%V4V8wd \1V!pV#uupiuupi)Fr)rlrmrr8TR14r7rJrrSrTr_r"rUrrrrorrWrrqrinsertr6r )r/rsrbrrr<processr:rwrYrrr;sinotherrpruABr@rfremrs& r0r;TR14..fsyyyI $$&DA~AU+AU+9 B Axxx}}  LLOEEAA! #s3:LLOLL&HA" NNA{{A"8 9#(ww'(U&*%(^3"1aBg AAwAJQ4>>>adnnntqt|R5AbE> ' AA#&qtQqT?D !tt|59&:TqttT&: #A$ 'q# 6!"159&:TqttT&: #A$ 'q# 6)!A$44$'$'!LL1Q4%!*Qqtyy|_a5G*G$)NO$qTQqT\tqt|R5AbE> ' AA#$Q4D)!A$44$'$'!LL1Q4%!*Qqtyy|_a5G*G$)NO$ LL1qt $ u: eB E';';s S #Srrs&f r0rGrGs,^@ R r2c*aaVV3Rlp\W4#)zConvert sin(x)**-2 to 1 + cot(x)**2. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR15 >>> from sympy.abc import x >>> from sympy import sin >>> TR15(1 - 1/sin(x)**2) -cot(x)**2 c j<\V\4'd!\VP\4'gV#VPpV^,^8Xd5\ VPV^,,4VP, #^V, p\ V\\RSSR7pW28wdTpV#)rFc^V,#rrrs&r0r!TR15..f..b!a%r2r)r6rrKrrTR15rr r/rYiar:rrs& r0r;TR15..fY2s## 277C(@(@I FF q5A:!a%()"''1 1 rT "c3Sc B 7B r2rr/rrr;s&ff r0rUrUI  R r2c*aaVV3Rlp\W4#)zConvert cos(x)**-2 to 1 + tan(x)**2. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR16 >>> from sympy.abc import x >>> from sympy import cos >>> TR16(1 - 1/cos(x)**2) -tan(x)**2 c j<\V\4'd!\VP\4'gV#VPpV^,^8Xd5\ VPV^,,4VP, #^V, p\ V\\RSSR7pW28wdTpV#)rFc^V,#rrrs&r0r!TR16..f..rTr2r)r6rrKrrrUrrrVs& r0r;TR16..fzrYr2rrZs&ff r0TR16rajr[r2cRp\W4#)a Convert f(x)**-i to g(x)**i where either ``i`` is an integer or the base is positive and f, g are: tan, cot; sin, csc; or cos, sec. Examples ======== >>> from sympy.simplify.fu import TR111 >>> from sympy.abc import x >>> from sympy import tan >>> TR111(1 - 1/tan(x)**2) 1 - cot(x)**2 c\V\4'dUVPP'g;VPP 'dVPP 'gV#\VP\4'd9\VPP^,4VP),#\VP\4'd9\VPP^,4VP),#\VP\4'd9\VPP^,4VP),#V#r5)r6rrKrTrrS is_negativerr r7rr"rr!r.s&r0r;TR111..fs r3   WW BFF$5$5$5"&&:L:L:LI bggs # #rww||A'"&&0 0  % %rww||A'"&&0 0  % %rww||A'"&&0 0 r2rr=s& r0TR111rfs  R r2c*aaVV3Rlp\W4#)a<Convert tan(x)**2 to sec(x)**2 - 1 and cot(x)**2 to csc(x)**2 - 1. See _TR56 docstring for advanced use of ``max`` and ``pow``. Examples ======== >>> from sympy.simplify.fu import TR22 >>> from sympy.abc import x >>> from sympy import tan, cot >>> TR22(1 + tan(x)**2) sec(x)**2 >>> TR22(1 + cot(x)**2) csc(x)**2 c <\V\4'd&VPP\\ 39gV#\ V\ \RSSR7p\ V\\RSSR7pV#)cV^, #rrrs&r0r!TR22..f..1q5r2rcV^, #rrrs&r0rrjrkr2) r6rrKrUr rrr!r"rs&r0r;TR22..fsU2s## c (BI 2sCcs C 2sCcs C r2rrZs&ff r0TR22rns$ R r2cRp\W4#)aConvert sin(x)**n and cos(x)**n with positive n to sums. Examples ======== >>> from sympy.simplify.fu import TRpower >>> from sympy.abc import x >>> from sympy import cos, sin >>> TRpower(sin(x)**6) -15*cos(2*x)/32 + 3*cos(4*x)/16 - cos(6*x)/32 + 5/16 >>> TRpower(sin(x)**3*cos(2*x)**4) (3*sin(x)/4 - sin(3*x)/4)*(cos(4*x)/2 + cos(8*x)/8 + 3/8) References ========== .. [1] https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulae c \V\4'd'\VP\\34'gV#VP 4wrVP ^,pVP'Ed6VP'Ed#VP'd\V\4'dv^^V, ,\\V^,^, 4Uu.uF3p\W$4\ V^V,, V,4,NK5 up!,pEMKVP'd\V\4'd^^V, ,\PV^, ^, ,,\\V^,^, 4Uu.uFOp\W$4\PV,,\V^V,, V,4,NKQ up!,pEMhVP'd\V\4'dn^^V, ,\\V^, 4Uu.uF3p\W$4\ V^V,, V,4,NK5 up!,pMVP'd\V\4'd^^V, ,\PV^, ,,\\V^, 4Uu.uFOp\W$4\PV,,\ V^V,, V,4,NKQ up!,pVP'd)V^V),\W"^, 4,, pV#uupiuupiuupiuupir5)r6rrKrrrr7rrTis_oddrrrrr#is_even)r/rwrsrrXs& r0r;TRpower..fs2s## 277S#J(G(GI~~ FF1I <<Fa^MM1$>%%(!ac'1%5>6>6?O=Q8RRz!S111Xc"1Q3Z$)'%-QN3AaC{3C$C$C'$)**z!S111Xammac223?DQqSz9K?I!:B!MM1$:%%(!ac'1%5:6:6?I9K4LLyyya1"ghqA#... $/=Q$)9Ks!9N. (AN3 9N8 AN= rr=s& r0TRpowerrts*, R r2c>\VP\44#)zReturn count of trigonometric functions in expression. Examples ======== >>> from sympy.simplify.fu import L >>> from sympy.abc import x >>> from sympy import cos, sin >>> L(cos(x)+sin(x)) 2 )rcountr$r.s&r0Lrws RXX+ , --r2c8\V4VP43#rI)rw count_opsrs&r0rr*sadAKKM2r2c \\V4p\\V4pTp\V4p\ V\ 4'g5VP !VPUu.uFp\WQR7NK up!#\V4pVP\\4'dIV!V4pV!V4V!V48dTpVP\\4'd \V4pVP\\4'd,V!V4p\!\#V44p\%W@Wg.VR7p\%\'V4WR7#uupi)aAttempt to simplify expression by using transformation rules given in the algorithm by Fu et al. :func:`fu` will try to minimize the objective function ``measure``. By default this first minimizes the number of trig terms and then minimizes the number of total operations. Examples ======== >>> from sympy.simplify.fu import fu >>> from sympy import cos, sin, tan, pi, S, sqrt >>> from sympy.abc import x, y, a, b >>> fu(sin(50)**2 + cos(50)**2 + sin(pi/6)) 3/2 >>> fu(sqrt(6)*cos(x) + sqrt(2)*sin(x)) 2*sqrt(2)*sin(x + pi/3) CTR1 example >>> eq = sin(x)**4 - cos(y)**2 + sin(y)**2 + 2*cos(x)**2 >>> fu(eq) cos(x)**4 - 2*cos(y)**2 + 2 CTR2 example >>> fu(S.Half - cos(2*x)/2) sin(x)**2 CTR3 example >>> fu(sin(a)*(cos(b) - sin(b)) + cos(a)*(sin(b) + cos(b))) sqrt(2)*sin(a + b + pi/4) CTR4 example >>> fu(sqrt(3)*cos(x)/2 + sin(x)/2) sin(x + pi/3) Example 1 >>> fu(1-sin(2*x)**2/4-sin(y)**2-cos(x)**4) -cos(x)**2 + cos(y)**2 Example 2 >>> fu(cos(4*pi/9)) sin(pi/18) >>> fu(cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9)) 1/16 Example 3 >>> fu(tan(7*pi/18)+tan(5*pi/18)-sqrt(3)*tan(5*pi/18)*tan(7*pi/18)) -sqrt(3) Objective function example >>> fu(sin(x)/cos(x)) # default objective function tan(x) >>> fu(sin(x)/cos(x), measure=lambda x: -x.count_ops()) # maximize op count sin(x)/cos(x) References ========== .. [1] https://www.sciencedirect.com/science/article/pii/S0895717706001609 )measure)rB)r'RL1RL2rr6rrUr7fur>hasrr rBrrrrDrry)r/r{fRL1fRL2rr:rv1rv2s&& r0r~r~*sL #w D #w D C B b$  wwAAA/ABB RB vvc32h CL72; &B 66#s  RB vvc32h(3-  #3$' 2 tBx ))BsE c\\4pV'd]VPFKpVP4wreV^8dV)pV)pYFV'd V!V4M^3,P V4KM MUV'dCVPF1pV\ P V!V43,P V4K3 M \R4h.pRpVFsp WI,p V wrk\V 4^8d7\V RR/p V!V 4p W8wdT p RpVP Wl,4KUVP Wj^,,4Ku V'd \V!pV#)zApply ``do`` to addends of ``rv`` that (if ``key1=True``) share at least a common absolute value of their coefficient and the value of ``key2`` when applied to the argument. If ``key1`` is False ``key2`` must be supplied and will be the only key applied. zmust have at least one keyFrkT) rror7rr_rr8 ValueErrorrWr)r/rkey2key1abscr:rr7rrXrgr,rYrs&&&& r0rrs( t D A>>#DA1uBB T!W!, - 4 4Q 7  A !%%a! " ) )! ,566 D C  G q6A:Q''AQ%Cx KK  KKA$  $Z Ir2z~ TR0 TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8 TR9 TR10 TR10i TR11 TR12 TR13 L TR2i TRmorrie TR12i TR14 TR15 TR16 TR111 TR22c\^4#rr#rr2r0_ROOT2r 7Nr2c\^4#)rrrr2r0rrrr2c&^\^4, #rrrr2r0rrs T!W9r2c aW3Uu.uFp\V4NK upwrVPV4wrEVPV4P4p^;rx\P VP 9d$VP\P 4pV)pMA\P VP 9d#VP\P 4pV)pWE3Uu.uFq3P4NK upwrRp V !W4p V fR#V wrp V !W4p V fR#V wrpV 'g V'gV 'd$\V \4'dWVWV 3wrrppYrV'gnT ;'gT pT;'gTp\VVP4'gR#WgVVP^,VP^,\V\43#V 'gV'gV 'dV'dV 'dV'd\WP4\VVP4JdR#W3Uu0uFpVPkK upo\;QJd!V3RlVV34F 'dK RM RM!V3RlVV344'gR#WgWP^,V P^,\WP43#V 'd V 'g)V'd V'gV'dV fV e VfVfR#T ;'gT pT;'gTpVPVP8wdR#V 'g\Pp V'g\PpWJd5V\4,pWgVVP^,\ ^, R3#W, \#48Xd4V^V,,pWgVVP^,\ ^, R3#W, \%48Xd4V^V ,,pWgVVP^,\ ^, R3#R#uupiuupiuupi)aReturn the gcd, s1, s2, a1, a2, bool where If two is False (default) then:: a + b = gcd*(s1*f(a1) + s2*f(a2)) where f = cos if bool else sin else: if bool, a + b was +/- cos(a1)*cos(a2) +/- sin(a1)*sin(a2) and equals n1*gcd*cos(a - b) if n1 == n2 else n1*gcd*cos(a + b) else a + b was +/- cos(a1)*sin(a2) +/- sin(a1)*cos(a2) and equals n1*gcd*sin(a + b) if n1 = n2 else n1*gcd*sin(b - a) Examples ======== >>> from sympy.simplify.fu import trig_split >>> from sympy.abc import x, y, z >>> from sympy import cos, sin, sqrt >>> trig_split(cos(x), cos(y)) (1, 1, 1, x, y, True) >>> trig_split(2*cos(x), -2*cos(y)) (2, 1, -1, x, y, True) >>> trig_split(cos(x)*sin(y), cos(y)*sin(y)) (sin(y), 1, 1, x, y, True) >>> trig_split(cos(x), -sqrt(3)*sin(x), two=True) (2, 1, -1, x, pi/6, False) >>> trig_split(cos(x), sin(x), two=True) (sqrt(2), 1, 1, x, pi/4, False) >>> trig_split(cos(x), -sin(x), two=True) (sqrt(2), 1, -1, x, pi/4, False) >>> trig_split(sqrt(2)*cos(x), -sqrt(6)*sin(x), two=True) (2*sqrt(2), 1, -1, x, pi/6, False) >>> trig_split(-sqrt(6)*cos(x), -sqrt(2)*sin(x), two=True) (-2*sqrt(2), 1, 1, x, pi/3, False) >>> trig_split(cos(x)/sqrt(6), sin(x)/sqrt(2), two=True) (sqrt(6)/3, 1, 1, x, pi/6, False) >>> trig_split(-sqrt(6)*cos(x)*sin(y), -sqrt(2)*sin(x)*sin(y), two=True) (-2*sqrt(2)*sin(y), 1, 1, x, pi/3, False) >>> trig_split(cos(x), sin(x)) >>> trig_split(cos(x), sin(z)) >>> trig_split(2*cos(x), -sin(x)) >>> trig_split(cos(x), -sqrt(3)*sin(x)) >>> trig_split(cos(x)*cos(y), sin(x)*sin(z)) >>> trig_split(cos(x)*cos(y), sin(x)*sin(y)) >>> trig_split(-sqrt(6)*cos(x), sqrt(2)*sin(x)*sin(y), two=True) cR;r#\PpVP'EdVP4wr@\ VP 4^8g V'gR#VP'd\ VP 4pMV.pVP^4p\V\4'dTpMT\V\4'dTpM;VP'd(VP\PJd W@,pMR#V'dV^,p\V\4'dV'dTpMbTpM_\V\4'dV'dTpM>TpM;VP'd(VP\PJd WF,pMR#V\PJdWBV3#RW#3#\V\4'dTpM\V\4'dTpVfVfR#V\PJdTMRpWBV3#)aEReturn ``a`` as a tuple (r, c, s) such that ``a = (r or 1)*(c or 1)*(s or 1)``. Three arguments are returned (radical, c-factor, s-factor) as long as the conditions set by ``two`` are met; otherwise None is returned. If ``two`` is True there will be one or two non-None values in the tuple: c and s or c and r or s and r or s or c with c being a cosine function (if possible) else a sine, and s being a sine function (if possible) else oosine. If ``two`` is False then there will only be a c or s term in the tuple. ``two`` also require that either two cos and/or sin be present (with the condition that if the functions are the same the arguments are different or vice versa) or that a single cosine or a single sine be present with an optional radical. If the above conditions dictated by ``two`` are not met then None is returned. N)rr8rlrrWr7rorqr6rrrJrr)r:rrrrr7rws&& r0 pow_cos_sintrig_split..pow_cos_sinsx(  UU 888NN$EB166{QcxxxAFF|s A!S!!As##aeeqvvoGa%%3''XXX!%%166/GB1552q8 8dA8 8 3  A 3  A 9 QUU?Raxr2Nc3@<"TFqPS9xK R#5irI)r7)rLrfr7s& r0rNtrig_split..^s<8a66T>8sFT)rr,ras_exprrr#factorsquor6rrUr7rr$r8rr rr)r:rwrrfuaubrrrrrcoacasacobcbsbrrrr7s&&& @r0rrs*d"# '1GAJ 'DA XXa[FB %%(   CKB}} " VVAMM "S "** $ VVAMM "S"$ *AIIK *DA?D AAyKCRAAyKCR B"B!4!4#&B#; "bB  HH" HH"!QVV$$AFF1Iqvvay*Q2DDD3rbRb''**R2II)+111s>> from sympy.simplify.fu import as_f_sign_1 >>> from sympy.abc import x >>> as_f_sign_1(x + 1) (1, x, 1) >>> as_f_sign_1(x - 1) (1, x, -1) >>> as_f_sign_1(-x + 1) (-1, x, -1) >>> as_f_sign_1(-x - 1) (-1, x, 1) >>> as_f_sign_1(2*x + 2) (2, x, 1) N)rVrWr7rr#r8rlrrr,rrrr) rYr:rwrrfrrrrrs & r0r"r"ws* 888s166{a' 66DAQ]]AEE "" EE 888q +++q A 2rqAQw ! '1GAJ 'DA XXa[FB %%(   C}} " VVAMM "   "** $ VVAMM "   "$ *AIIK *DAAEEz1B RxdSAEEzrz+ ( +s H2Hc&aV3Rlp\W4#)aReplace all hyperbolic functions with trig functions using the Osborne rule. Notes ===== ``d`` is a dummy variable to prevent automatic evaluation of trigonometric/hyperbolic functions. References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function c"<\V\4'gV#VP^,pVP'g VS,M5\P !VPUu.uF q"S,NK up4p\V\ 4'd\\V4,#\V\4'd \V4#\V\4'd\\V4,#\V\4'd\V4\, #\V\4'd \!V4#\V\"4'd\%V4\, #\'RVP(,4huupi)r unhandled %s)r6rr7rVrrrr rrrrrrr rr!rr"NotImplementedErrorrU)r/r:rfrbs& r0r;_osborne..fs"011I GGAJxxxAaCS^^!&&4I&QqSS&4I%J b$  SV8O D ! !q6M D ! !SV8O D ! !q6!8O D ! !q6M D ! !q6!8O%nrww&>? ?5Js&F rrYrbr;s&f r0_osborners"@( Q?r2c&aV3Rlp\W4#)a Replace all trig functions with hyperbolic functions using the Osborne rule. Notes ===== ``d`` is a dummy variable to prevent automatic evaluation of trigonometric/hyperbolic functions. References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function c<\V\4'gV#VP^,PSRR7wrVP S\ P /4V\,,p\V\4'd\V4\, #\V\4'd \V4#\V\4'd\V4\, #\V\4'd\V4\,#\V\ 4'd \#V4#\V\$4'd\'V4\,#\)RVP*,4h)rT)as_Addr)r6r$r7as_independentxreplacerr8r rrrrrrr rr!rr"rrrU)r/constrr:rbs& r0r;_osbornei..fs"344I771:,,Qt,< JJ155z "U1W , b#  719  C 7N C 719  C 719  C 7N C 719 %nrww&>? ?r2rrs&f r0 _osborneirs @( Q?r2c8aaaa ^RIHo ^RIHoVP \ 4pVUu.uFq"\ 43NK upoVP\S44pSUUu.uFwrEWT3NK uppo\ 4o\VS4VVVV 3Rl3#uupiuuppi)aReturn an expression containing hyperbolic functions in terms of trigonometric functions. Any trigonometric functions initially present are replaced with Dummy symbols and the function to undo the masking and the conversion back to hyperbolics is also returned. It should always be true that:: t, f = hyper_as_trig(expr) expr == f(t) Examples ======== >>> from sympy.simplify.fu import hyper_as_trig, fu >>> from sympy.abc import x >>> from sympy import cosh, sinh >>> eq = sinh(x)**2 + cosh(x)**2 >>> t, f = hyper_as_trig(eq) >>> f(fu(t)) cosh(2*x) References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function r{)collectc <S!S!\VS4P\S444\P4#rI)rrdictr ImaginaryUnit)rrrbrepsr|s&r0rhyper_as_trig..&s1'(!Q  d,3./0+@r2) rr|sympy.simplify.radsimpratomsr$rrrr) r/trigsrumaskedrXrgrrbrr|s & @@@@r0 hyper_as_trigrs41. HH* +E"' (%QL% (D [[d $F $ $ttqQFt $D A FA !@ @@ ) %s B BcVP\\4'gV#\\ \ V444#)aConvert products and powers of sin and cos to sums. Explanation =========== Applied power reduction TRpower first, then expands products, and converts products to sums with TR8. Examples ======== >>> from sympy.simplify.fu import sincos_to_sum >>> from sympy.abc import x >>> from sympy import cos, sin >>> sincos_to_sum(16*sin(x)**3*cos(2*x)**2) 7*sin(x) - 5*sin(3*x) + 3*sin(5*x) - sin(7*x) )rrrrr rt)exprs&r0 sincos_to_sumr*s.& 88C   :gdm,--r2)F)rFrrI)NT)r collectionsrsympy.core.addrsympy.core.cachersympy.core.exprrsympy.core.exprtoolsrrr sympy.core.functionr sympy.core.mulr sympy.core.numbersr r sympy.core.powerrsympy.core.singletonrsympy.core.sortingrsympy.core.symbolrsympy.core.sympifyrsympy.core.traversalr(sympy.functions.combinatorial.factorialsr%sympy.functions.elementary.hyperbolicrrrrrrr(sympy.functions.elementary.trigonometricrrrr r!r"r#r$sympy.ntheory.factor_r%sympy.polys.polytoolsr&sympy.strategies.treer'sympy.strategies.corer(r)sympyr*r1r>rBryrrrrrrrrrrrrrr1r7rDrGrUrarfrnrtrwrorCTR1CTR2CTR3CTR4r|r}r~rrfufuncsrziplocalsgetFUrrrrr"rrrrrr2r0rs#$ AA*$ "&#&*=<<<???/((1 )0?      o7d6r%P$N(@V.r2