4.21

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.21.E1	${\displaystyle{\displaystyle\sin u+\cos u=\sqrt{2}\sin\left(u+\tfrac{1}{4}\pi\right)}}$ \sin@@{u}+\cos@@{u} = \sqrt{2}\sin@{u+\tfrac{1}{4}\pi}		sin(u)+ cos(u) = sqrt(2)*sin(u +(1)/(4)*Pi)	Sin[u]+ Cos[u] == Sqrt[2]*Sin[u +Divide[1,4]*Pi]	Successful	Successful	-	Successful [Tested: 10]
4.21.E1	${\displaystyle{\displaystyle\sin u-\cos u=\sqrt{2}\sin\left(u-\tfrac{1}{4}\pi\right)}}$ \sin@@{u}-\cos@@{u} = \sqrt{2}\sin@{u-\tfrac{1}{4}\pi}		sin(u)- cos(u) = sqrt(2)*sin(u -(1)/(4)*Pi)	Sin[u]- Cos[u] == Sqrt[2]*Sin[u -Divide[1,4]*Pi]	Successful	Successful	-	Successful [Tested: 10]
4.21.E1	${\displaystyle{\displaystyle\sqrt{2}\sin\left(u+\tfrac{1}{4}\pi\right)=+\sqrt{2}\cos\left(u-\tfrac{1}{4}\pi\right)}}$ \sqrt{2}\sin@{u+\tfrac{1}{4}\pi} = +\sqrt{2}\cos@{u-\tfrac{1}{4}\pi}		sqrt(2)*sin(u +(1)/(4)*Pi) = +sqrt(2)*cos(u -(1)/(4)*Pi)	Sqrt[2]*Sin[u +Divide[1,4]*Pi] == +Sqrt[2]*Cos[u -Divide[1,4]*Pi]	Successful	Successful	-	Successful [Tested: 10]
4.21.E1	${\displaystyle{\displaystyle\sqrt{2}\sin\left(u-\tfrac{1}{4}\pi\right)=-\sqrt{2}\cos\left(u+\tfrac{1}{4}\pi\right)}}$ \sqrt{2}\sin@{u-\tfrac{1}{4}\pi} = -\sqrt{2}\cos@{u+\tfrac{1}{4}\pi}		sqrt(2)*sin(u -(1)/(4)*Pi) = -sqrt(2)*cos(u +(1)/(4)*Pi)	Sqrt[2]*Sin[u -Divide[1,4]*Pi] == -Sqrt[2]*Cos[u +Divide[1,4]*Pi]	Successful	Successful	-	Successful [Tested: 10]
4.21.E2	${\displaystyle{\displaystyle\sin\left(u+v\right)=\sin u\cos v+\cos u\sin v}}$ \sin@{u+ v} = \sin@@{u}\cos@@{v}+\cos@@{u}\sin@@{v}		sin(u + v) = sin(u)*cos(v)+ cos(u)*sin(v)	Sin[u + v] == Sin[u]*Cos[v]+ Cos[u]*Sin[v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E2	${\displaystyle{\displaystyle\sin\left(u-v\right)=\sin u\cos v-\cos u\sin v}}$ \sin@{u- v} = \sin@@{u}\cos@@{v}-\cos@@{u}\sin@@{v}		sin(u - v) = sin(u)*cos(v)- cos(u)*sin(v)	Sin[u - v] == Sin[u]*Cos[v]- Cos[u]*Sin[v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E3	${\displaystyle{\displaystyle\cos\left(u+v\right)=\cos u\cos v-\sin u\sin v}}$ \cos@{u+ v} = \cos@@{u}\cos@@{v}-\sin@@{u}\sin@@{v}		cos(u + v) = cos(u)*cos(v)- sin(u)*sin(v)	Cos[u + v] == Cos[u]*Cos[v]- Sin[u]*Sin[v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E3	${\displaystyle{\displaystyle\cos\left(u-v\right)=\cos u\cos v+\sin u\sin v}}$ \cos@{u- v} = \cos@@{u}\cos@@{v}+\sin@@{u}\sin@@{v}		cos(u - v) = cos(u)*cos(v)+ sin(u)*sin(v)	Cos[u - v] == Cos[u]*Cos[v]+ Sin[u]*Sin[v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E4	${\displaystyle{\displaystyle\tan\left(u+v\right)=\frac{\tan u+\tan v}{1-\tan u\tan v}}}$ \tan@{u+ v} = \frac{\tan@@{u}+\tan@@{v}}{1-\tan@@{u}\tan@@{v}}		tan(u + v) = (tan(u)+ tan(v))/(1 - tan(u)*tan(v))	Tan[u + v] == Divide[Tan[u]+ Tan[v],1 - Tan[u]*Tan[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E4	${\displaystyle{\displaystyle\tan\left(u-v\right)=\frac{\tan u-\tan v}{1+\tan u\tan v}}}$ \tan@{u- v} = \frac{\tan@@{u}-\tan@@{v}}{1+\tan@@{u}\tan@@{v}}		tan(u - v) = (tan(u)- tan(v))/(1 + tan(u)*tan(v))	Tan[u - v] == Divide[Tan[u]- Tan[v],1 + Tan[u]*Tan[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E5	${\displaystyle{\displaystyle\cot\left(u+v\right)=\frac{+\cot u\cot v-1}{\cot u+\cot v}}}$ \cot@{u+ v} = \frac{+\cot@@{u}\cot@@{v}-1}{\cot@@{u}+\cot@@{v}}		cot(u + v) = (+ cot(u)*cot(v)- 1)/(cot(u)+ cot(v))	Cot[u + v] == Divide[+ Cot[u]*Cot[v]- 1,Cot[u]+ Cot[v]]	Successful	Successful	-	Failed [10 / 100] Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: Complex[1.9674787081851645*^15, 2.0439439417914815*^15] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.21.E5	${\displaystyle{\displaystyle\cot\left(u-v\right)=\frac{-\cot u\cot v-1}{\cot u-\cot v}}}$ \cot@{u- v} = \frac{-\cot@@{u}\cot@@{v}-1}{\cot@@{u}-\cot@@{v}}		cot(u - v) = (- cot(u)*cot(v)- 1)/(cot(u)- cot(v))	Cot[u - v] == Divide[- Cot[u]*Cot[v]- 1,Cot[u]- Cot[v]]	Successful	Successful	-	Failed [10 / 100] Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.21.E6	${\displaystyle{\displaystyle\sin u+\sin v=2\sin\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)}}$ \sin@@{u}+\sin@@{v} = 2\sin@{\frac{u+v}{2}}\cos@{\frac{u-v}{2}}		sin(u)+ sin(v) = 2*sin((u + v)/(2))*cos((u - v)/(2))	Sin[u]+ Sin[v] == 2*Sin[Divide[u + v,2]]*Cos[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E7	${\displaystyle{\displaystyle\sin u-\sin v=2\cos\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)}}$ \sin@@{u}-\sin@@{v} = 2\cos@{\frac{u+v}{2}}\sin@{\frac{u-v}{2}}		sin(u)- sin(v) = 2*cos((u + v)/(2))*sin((u - v)/(2))	Sin[u]- Sin[v] == 2*Cos[Divide[u + v,2]]*Sin[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E8	${\displaystyle{\displaystyle\cos u+\cos v=2\cos\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)}}$ \cos@@{u}+\cos@@{v} = 2\cos@{\frac{u+v}{2}}\cos@{\frac{u-v}{2}}		cos(u)+ cos(v) = 2*cos((u + v)/(2))*cos((u - v)/(2))	Cos[u]+ Cos[v] == 2*Cos[Divide[u + v,2]]*Cos[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E9	${\displaystyle{\displaystyle\cos u-\cos v=-2\sin\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)}}$ \cos@@{u}-\cos@@{v} = -2\sin@{\frac{u+v}{2}}\sin@{\frac{u-v}{2}}		cos(u)- cos(v) = - 2*sin((u + v)/(2))*sin((u - v)/(2))	Cos[u]- Cos[v] == - 2*Sin[Divide[u + v,2]]*Sin[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E10	${\displaystyle{\displaystyle\tan u+\tan v=\frac{\sin\left(u+v\right)}{\cos u\cos v}}}$ \tan@@{u}+\tan@@{v} = \frac{\sin@{u+ v}}{\cos@@{u}\cos@@{v}}		tan(u)+ tan(v) = (sin(u + v))/(cos(u)*cos(v))	Tan[u]+ Tan[v] == Divide[Sin[u + v],Cos[u]*Cos[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E10	${\displaystyle{\displaystyle\tan u-\tan v=\frac{\sin\left(u-v\right)}{\cos u\cos v}}}$ \tan@@{u}-\tan@@{v} = \frac{\sin@{u- v}}{\cos@@{u}\cos@@{v}}		tan(u)- tan(v) = (sin(u - v))/(cos(u)*cos(v))	Tan[u]- Tan[v] == Divide[Sin[u - v],Cos[u]*Cos[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E11	${\displaystyle{\displaystyle\cot u+\cot v=\frac{\sin\left(v+u\right)}{\sin u\sin v}}}$ \cot@@{u}+\cot@@{v} = \frac{\sin@{v+ u}}{\sin@@{u}\sin@@{v}}		cot(u)+ cot(v) = (sin(v + u))/(sin(u)*sin(v))	Cot[u]+ Cot[v] == Divide[Sin[v + u],Sin[u]*Sin[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E11	${\displaystyle{\displaystyle\cot u-\cot v=\frac{\sin\left(v-u\right)}{\sin u\sin v}}}$ \cot@@{u}-\cot@@{v} = \frac{\sin@{v- u}}{\sin@@{u}\sin@@{v}}		cot(u)- cot(v) = (sin(v - u))/(sin(u)*sin(v))	Cot[u]- Cot[v] == Divide[Sin[v - u],Sin[u]*Sin[v]]	Successful	Successful	-	Successful [Tested: 100]
4.21.E12	${\displaystyle{\displaystyle{\sin^{2}}z+{\cos^{2}}z=1}}$ \sin^{2}@@{z}+\cos^{2}@@{z} = 1		(sin(z))^(2)+ (cos(z))^(2) = 1	(Sin[z])^(2)+ (Cos[z])^(2) == 1	Successful	Successful	-	Successful [Tested: 7]
4.21.E13	${\displaystyle{\displaystyle{\sec^{2}}z=1+{\tan^{2}}z}}$ \sec^{2}@@{z} = 1+\tan^{2}@@{z}		(sec(z))^(2) = 1 + (tan(z))^(2)	(Sec[z])^(2) == 1 + (Tan[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.21.E14	${\displaystyle{\displaystyle{\csc^{2}}z=1+{\cot^{2}}z}}$ \csc^{2}@@{z} = 1+\cot^{2}@@{z}		(csc(z))^(2) = 1 + (cot(z))^(2)	(Csc[z])^(2) == 1 + (Cot[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.21.E15	${\displaystyle{\displaystyle 2\sin u\sin v=\cos\left(u-v\right)-\cos\left(u+v\right)}}$ 2\sin@@{u}\sin@@{v} = \cos@{u-v}-\cos@{u+v}		2*sin(u)*sin(v) = cos(u - v)- cos(u + v)	2*Sin[u]*Sin[v] == Cos[u - v]- Cos[u + v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E16	${\displaystyle{\displaystyle 2\cos u\cos v=\cos\left(u-v\right)+\cos\left(u+v\right)}}$ 2\cos@@{u}\cos@@{v} = \cos@{u-v}+\cos@{u+v}		2*cos(u)*cos(v) = cos(u - v)+ cos(u + v)	2*Cos[u]*Cos[v] == Cos[u - v]+ Cos[u + v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E17	${\displaystyle{\displaystyle 2\sin u\cos v=\sin\left(u-v\right)+\sin\left(u+v\right)}}$ 2\sin@@{u}\cos@@{v} = \sin@{u-v}+\sin@{u+v}		2*sin(u)*cos(v) = sin(u - v)+ sin(u + v)	2*Sin[u]*Cos[v] == Sin[u - v]+ Sin[u + v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E18	${\displaystyle{\displaystyle{\sin^{2}}u-{\sin^{2}}v=\sin\left(u+v\right)\sin\left(u-v\right)}}$ \sin^{2}@@{u}-\sin^{2}@@{v} = \sin@{u+v}\sin@{u-v}		(sin(u))^(2)- (sin(v))^(2) = sin(u + v)*sin(u - v)	(Sin[u])^(2)- (Sin[v])^(2) == Sin[u + v]*Sin[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E19	${\displaystyle{\displaystyle{\cos^{2}}u-{\cos^{2}}v=-\sin\left(u+v\right)\sin\left(u-v\right)}}$ \cos^{2}@@{u}-\cos^{2}@@{v} = -\sin@{u+v}\sin@{u-v}		(cos(u))^(2)- (cos(v))^(2) = - sin(u + v)*sin(u - v)	(Cos[u])^(2)- (Cos[v])^(2) == - Sin[u + v]*Sin[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E20	${\displaystyle{\displaystyle{\cos^{2}}u-{\sin^{2}}v=\cos\left(u+v\right)\cos\left(u-v\right)}}$ \cos^{2}@@{u}-\sin^{2}@@{v} = \cos@{u+v}\cos@{u-v}		(cos(u))^(2)- (sin(v))^(2) = cos(u + v)*cos(u - v)	(Cos[u])^(2)- (Sin[v])^(2) == Cos[u + v]*Cos[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.21.E21	${\displaystyle{\displaystyle\sin\frac{z}{2}=+\left(\frac{1-\cos z}{2}\right)^{1/2}}}$ \sin@@{\frac{z}{2}} = +\left(\frac{1-\cos@@{z}}{2}\right)^{1/2}		sin((z)/(2)) = +((1 - cos(z))/(2))^(1/2)	Sin[Divide[z,2]] == +(Divide[1 - Cos[z],2])^(1/2)	Failure	Failure	Failed [2 / 7] Result: -.5419255224+.8655716642*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: -.8655770340-.4585952894*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[-0.541925522457336, 0.8655716640572733] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8655770337160631, -0.4585952893468805] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.21.E21	${\displaystyle{\displaystyle\sin\frac{z}{2}=-\left(\frac{1-\cos z}{2}\right)^{1/2}}}$ \sin@@{\frac{z}{2}} = -\left(\frac{1-\cos@@{z}}{2}\right)^{1/2}		sin((z)/(2)) = -((1 - cos(z))/(2))^(1/2)	Sin[Divide[z,2]] == -(Divide[1 - Cos[z],2])^(1/2)	Failure	Failure	Failed [5 / 7] Result: .8655770340+.4585952894*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: .5419255224-.8655716642*I Test Values: {z = 1/2-1/2*I*3^(1/2)} Result: 1.363277520 Test Values: {z = 1.5} Result: .4948079184 Test Values: {z = .5} ... skip entries to safe data	Failed [5 / 7] Result: Complex[0.8655770337160631, 0.4585952893468805] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.5419255224573365, -0.8655716640572731] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.21.E22	${\displaystyle{\displaystyle\cos\frac{z}{2}=+\left(\frac{1+\cos z}{2}\right)^{1/2}}}$ \cos@@{\frac{z}{2}} = +\left(\frac{1+\cos@@{z}}{2}\right)^{1/2}		cos((z)/(2)) = +((1 + cos(z))/(2))^(1/2)	Cos[Divide[z,2]] == +(Divide[1 + Cos[z],2])^(1/2)	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.21.E22	${\displaystyle{\displaystyle\cos\frac{z}{2}=-\left(\frac{1+\cos z}{2}\right)^{1/2}}}$ \cos@@{\frac{z}{2}} = -\left(\frac{1+\cos@@{z}}{2}\right)^{1/2}		cos((z)/(2)) = -((1 + cos(z))/(2))^(1/2)	Cos[Divide[z,2]] == -(Divide[1 + Cos[z],2])^(1/2)	Failure	Failure	Failed [7 / 7] Result: 1.872439139-.2119959694*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: 2.122352334+.2210167318*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: 2.122352334+.2210167318*I Test Values: {z = 1/2-1/2*I*3^(1/2)} Result: 1.872439139-.2119959694*I Test Values: {z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [7 / 7] Result: Complex[1.872439138961815, -0.2119959693051084] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[2.1223523339444896, 0.22101673165487346] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.21.E23	${\displaystyle{\displaystyle\tan\frac{z}{2}=+\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}}}$ \tan@@{\frac{z}{2}} = +\left(\frac{1-\cos@@{z}}{1+\cos@@{z}}\right)^{1/2}		tan((z)/(2)) = +((1 - cos(z))/(1 + cos(z)))^(1/2)	Tan[Divide[z,2]] == +(Divide[1 - Cos[z],1 + Cos[z]])^(1/2)	Failure	Failure	Failed [2 / 7] Result: -.4211742148+.8595320616*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: -.8580864930-.5869891489*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[-0.4211742148849969, 0.8595320613685856] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.858086492859854, -0.5869891488727426] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.21.E23	${\displaystyle{\displaystyle\tan\frac{z}{2}=-\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}}}$ \tan@@{\frac{z}{2}} = -\left(\frac{1-\cos@@{z}}{1+\cos@@{z}}\right)^{1/2}		tan((z)/(2)) = -((1 - cos(z))/(1 + cos(z)))^(1/2)	Tan[Divide[z,2]] == -(Divide[1 - Cos[z],1 + Cos[z]])^(1/2)	Failure	Failure	Failed [5 / 7] Result: .8580864930+.5869891489*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: .4211742148-.8595320616*I Test Values: {z = 1/2-1/2*I*3^(1/2)} Result: 1.863192920 Test Values: {z = 1.5} Result: .5106838424 Test Values: {z = .5} ... skip entries to safe data	Failed [5 / 7] Result: Complex[0.858086492859854, 0.5869891488727426] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.4211742148849973, -0.8595320613685857] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.21.E23	${\displaystyle{\displaystyle+\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=\frac{1-\cos z}{\sin z}}}$ +\left(\frac{1-\cos@@{z}}{1+\cos@@{z}}\right)^{1/2} = \frac{1-\cos@@{z}}{\sin@@{z}}		+((1 - cos(z))/(1 + cos(z)))^(1/2) = (1 - cos(z))/(sin(z))	+(Divide[1 - Cos[z],1 + Cos[z]])^(1/2) == Divide[1 - Cos[z],Sin[z]]	Failure	Failure	Failed [2 / 7] Result: .4211742148-.8595320615*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: .8580864930+.5869891489*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[0.42117421488499684, -0.8595320613685857] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.8580864928598539, 0.5869891488727426] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.21.E23	${\displaystyle{\displaystyle-\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=\frac{1-\cos z}{\sin z}}}$ -\left(\frac{1-\cos@@{z}}{1+\cos@@{z}}\right)^{1/2} = \frac{1-\cos@@{z}}{\sin@@{z}}		-((1 - cos(z))/(1 + cos(z)))^(1/2) = (1 - cos(z))/(sin(z))	-(Divide[1 - Cos[z],1 + Cos[z]])^(1/2) == Divide[1 - Cos[z],Sin[z]]	Failure	Failure	Failed [5 / 7] Result: -.8580864930-.5869891489*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -.4211742148+.8595320615*I Test Values: {z = 1/2-1/2*I*3^(1/2)} Result: -1.863192920 Test Values: {z = 1.5} Result: -.5106838424 Test Values: {z = .5} ... skip entries to safe data	Failed [5 / 7] Result: Complex[-0.8580864928598539, -0.5869891488727426] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.4211742148849972, 0.8595320613685855] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.21.E23	${\displaystyle{\displaystyle\frac{1-\cos z}{\sin z}=\frac{\sin z}{1+\cos z}}}$ \frac{1-\cos@@{z}}{\sin@@{z}} = \frac{\sin@@{z}}{1+\cos@@{z}}		(1 - cos(z))/(sin(z)) = (sin(z))/(1 + cos(z))	Divide[1 - Cos[z],Sin[z]] == Divide[Sin[z],1 + Cos[z]]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
4.21.E24	${\displaystyle{\displaystyle\sin\left(-z\right)=-\sin z}}$ \sin@{-z} = -\sin@@{z}		sin(- z) = - sin(z)	Sin[- z] == - Sin[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E25	${\displaystyle{\displaystyle\cos\left(-z\right)=\cos z}}$ \cos@{-z} = \cos@@{z}		cos(- z) = cos(z)	Cos[- z] == Cos[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E26	${\displaystyle{\displaystyle\tan\left(-z\right)=-\tan z}}$ \tan@{-z} = -\tan@@{z}		tan(- z) = - tan(z)	Tan[- z] == - Tan[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E27	${\displaystyle{\displaystyle\sin\left(2z\right)=2\sin z\cos z}}$ \sin@{2z} = 2\sin@@{z}\cos@@{z}		sin(2*z) = 2*sin(z)*cos(z)	Sin[2*z] == 2*Sin[z]*Cos[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E27	${\displaystyle{\displaystyle 2\sin z\cos z=\frac{2\tan z}{1+{\tan^{2}}z}}}$ 2\sin@@{z}\cos@@{z} = \frac{2\tan@@{z}}{1+\tan^{2}@@{z}}		2*sin(z)*cos(z) = (2*tan(z))/(1 + (tan(z))^(2))	2*Sin[z]*Cos[z] == Divide[2*Tan[z],1 + (Tan[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.21.E28	${\displaystyle{\displaystyle\cos\left(2z\right)=2{\cos^{2}}z-1}}$ \cos@{2z} = 2\cos^{2}@@{z}-1		cos(2*z) = 2*(cos(z))^(2)- 1	Cos[2*z] == 2*(Cos[z])^(2)- 1	Successful	Successful	-	Successful [Tested: 7]
4.21.E28	${\displaystyle{\displaystyle 2{\cos^{2}}z-1=1-2{\sin^{2}}z}}$ 2\cos^{2}@@{z}-1 = 1-2\sin^{2}@@{z}		2*(cos(z))^(2)- 1 = 1 - 2*(sin(z))^(2)	2*(Cos[z])^(2)- 1 == 1 - 2*(Sin[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.21.E28	${\displaystyle{\displaystyle 1-2{\sin^{2}}z={\cos^{2}}z-{\sin^{2}}z}}$ 1-2\sin^{2}@@{z} = \cos^{2}@@{z}-\sin^{2}@@{z}		1 - 2*(sin(z))^(2) = (cos(z))^(2)- (sin(z))^(2)	1 - 2*(Sin[z])^(2) == (Cos[z])^(2)- (Sin[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.21.E28	${\displaystyle{\displaystyle{\cos^{2}}z-{\sin^{2}}z=\frac{1-{\tan^{2}}z}{1+{\tan^{2}}z}}}$ \cos^{2}@@{z}-\sin^{2}@@{z} = \frac{1-\tan^{2}@@{z}}{1+\tan^{2}@@{z}}		(cos(z))^(2)- (sin(z))^(2) = (1 - (tan(z))^(2))/(1 + (tan(z))^(2))	(Cos[z])^(2)- (Sin[z])^(2) == Divide[1 - (Tan[z])^(2),1 + (Tan[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.21.E29	${\displaystyle{\displaystyle\tan\left(2z\right)=\frac{2\tan z}{1-{\tan^{2}}z}}}$ \tan@{2z} = \frac{2\tan@@{z}}{1-\tan^{2}@@{z}}		tan(2*z) = (2*tan(z))/(1 - (tan(z))^(2))	Tan[2*z] == Divide[2*Tan[z],1 - (Tan[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.21.E29	${\displaystyle{\displaystyle\frac{2\tan z}{1-{\tan^{2}}z}=\frac{2\cot z}{{\cot^{2}}z-1}}}$ \frac{2\tan@@{z}}{1-\tan^{2}@@{z}} = \frac{2\cot@@{z}}{\cot^{2}@@{z}-1}		(2*tan(z))/(1 - (tan(z))^(2)) = (2*cot(z))/((cot(z))^(2)- 1)	Divide[2*Tan[z],1 - (Tan[z])^(2)] == Divide[2*Cot[z],(Cot[z])^(2)- 1]	Successful	Successful	-	Successful [Tested: 7]
4.21.E29	${\displaystyle{\displaystyle\frac{2\cot z}{{\cot^{2}}z-1}=\frac{2}{\cot z-\tan z}}}$ \frac{2\cot@@{z}}{\cot^{2}@@{z}-1} = \frac{2}{\cot@@{z}-\tan@@{z}}		(2*cot(z))/((cot(z))^(2)- 1) = (2)/(cot(z)- tan(z))	Divide[2*Cot[z],(Cot[z])^(2)- 1] == Divide[2,Cot[z]- Tan[z]]	Successful	Successful	-	Successful [Tested: 7]
4.21.E30	${\displaystyle{\displaystyle\sin\left(3z\right)=3\sin z-4{\sin^{3}}z}}$ \sin@{3z} = 3\sin@@{z}-4\sin^{3}@@{z}		sin(3*z) = 3*sin(z)- 4*(sin(z))^(3)	Sin[3*z] == 3*Sin[z]- 4*(Sin[z])^(3)	Successful	Successful	-	Successful [Tested: 7]
4.21.E31	${\displaystyle{\displaystyle\cos\left(3z\right)=-3\cos z+4{\cos^{3}}z}}$ \cos@{3z} = -3\cos@@{z}+4\cos^{3}@@{z}		cos(3*z) = - 3*cos(z)+ 4*(cos(z))^(3)	Cos[3*z] == - 3*Cos[z]+ 4*(Cos[z])^(3)	Successful	Successful	-	Successful [Tested: 7]
4.21.E32	${\displaystyle{\displaystyle\sin\left(4z\right)=8{\cos^{3}}z\sin z-4\cos z\sin z}}$ \sin@{4z} = 8\cos^{3}@@{z}\sin@@{z}-4\cos@@{z}\sin@@{z}		sin(4*z) = 8*(cos(z))^(3)* sin(z)- 4*cos(z)*sin(z)	Sin[4*z] == 8*(Cos[z])^(3)* Sin[z]- 4*Cos[z]*Sin[z]	Successful	Successful	-	Successful [Tested: 7]
4.21.E33	${\displaystyle{\displaystyle\cos\left(4z\right)=8{\cos^{4}}z-8{\cos^{2}}z+1}}$ \cos@{4z} = 8\cos^{4}@@{z}-8\cos^{2}@@{z}+1		cos(4*z) = 8*(cos(z))^(4)- 8*(cos(z))^(2)+ 1	Cos[4*z] == 8*(Cos[z])^(4)- 8*(Cos[z])^(2)+ 1	Successful	Successful	-	Successful [Tested: 7]
4.21.E34	${\displaystyle{\displaystyle\cos\left(nz\right)+i\sin\left(nz\right)=(\cos z+i\sin z)^{n}}}$ \cos@{nz}+i\sin@{nz} = (\cos@@{z}+i\sin@@{z})^{n}		cos(n*z)+ I*sin(n*z) = (cos(z)+ I*sin(z))^(n)	Cos[n*z]+ I*Sin[n*z] == (Cos[z]+ I*Sin[z])^(n)	Successful	Failure	-	Successful [Tested: 21]
4.21.E35	${\displaystyle{\displaystyle\sin\left(nz\right)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(z+\frac{k\pi}{n}\right)}}$ \sin@{nz} = 2^{n-1}\prod_{k=0}^{n-1}\sin@{z+\frac{k\pi}{n}}		sin(n*z) = (2)^(n - 1)* product(sin(z +(k*Pi)/(n)), k = 0..n - 1)	Sin[n*z] == (2)^(n - 1)* Product[Sin[z +Divide[k*Pi,n]], {k, 0, n - 1}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 21]	Successful [Tested: 7]
4.21#Ex1	${\displaystyle{\displaystyle\sin z=\frac{2t}{1+t^{2}}}}$ \sin@@{z} = \frac{2t}{1+t^{2}}		sin(z) = (2*t)/(1 + (t)^(2))	Sin[z] == Divide[2*t,1 + (t)^(2)]	Failure	Failure	Failed [42 / 42] Result: 1.782057258+.3375964631*I Test Values: {t = -1.5, z = 1/2*3^(1/2)+1/2*I} Result: .2523455641+.8586367171*I Test Values: {t = -1.5, z = -1/2+1/2*I*3^(1/2)} Result: 1.593808282-.8586367171*I Test Values: {t = -1.5, z = 1/2-1/2*I*3^(1/2)} Result: .640965885e-1-.3375964631*I Test Values: {t = -1.5, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [42 / 42] Result: Complex[1.782057257377061, 0.33759646322287] Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.25234556426971166, 0.8586367168171449] Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.21#Ex2	${\displaystyle{\displaystyle\cos z=\frac{1-t^{2}}{1+t^{2}}}}$ \cos@@{z} = \frac{1-t^{2}}{1+t^{2}}		cos(z) = (1 - (t)^(2))/(1 + (t)^(2))	Cos[z] == Divide[1 - (t)^(2),1 + (t)^(2)]	Failure	Failure	Failed [42 / 42] Result: 1.115158404-.3969495503*I Test Values: {t = -1.5, z = 1/2*3^(1/2)+1/2*I} Result: 1.612380902+.4690753764*I Test Values: {t = -1.5, z = -1/2+1/2*I*3^(1/2)} Result: 1.612380902+.4690753764*I Test Values: {t = -1.5, z = 1/2-1/2*I*3^(1/2)} Result: 1.115158404-.3969495503*I Test Values: {t = -1.5, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [42 / 42] Result: Complex[1.1151584036726099, -0.3969495502290325] Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.612380901479495, 0.46907537626850365] Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.21.E37	${\displaystyle{\displaystyle\sin z=\sin x\cosh y+\mathrm{i}\cos x\sinh y}}$ \sin@@{z} = \sin@@{x}\cosh@@{y}+\iunit\cos@@{x}\sinh@@{y}		sin(x + y*I) = sin(x)*cosh(y)+ I*cos(x)*sinh(y)	Sin[x + y*I] == Sin[x]*Cosh[y]+ I*Cos[x]*Sinh[y]	Successful	Successful	-	Successful [Tested: 18]
4.21.E38	${\displaystyle{\displaystyle\cos z=\cos x\cosh y-\mathrm{i}\sin x\sinh y}}$ \cos@@{z} = \cos@@{x}\cosh@@{y}-\iunit\sin@@{x}\sinh@@{y}		cos(x + y*I) = cos(x)*cosh(y)- I*sin(x)*sinh(y)	Cos[x + y*I] == Cos[x]*Cosh[y]- I*Sin[x]*Sinh[y]	Successful	Successful	-	Successful [Tested: 18]
4.21.E39	${\displaystyle{\displaystyle\tan z=\frac{\sin\left(2x\right)+\mathrm{i}\sinh\left(2y\right)}{\cos\left(2x\right)+\cosh\left(2y\right)}}}$ \tan@@{z} = \frac{\sin@{2x}+\iunit\sinh@{2y}}{\cos@{2x}+\cosh@{2y}}		tan(x + y*I) = (sin(2*x)+ I*sinh(2*y))/(cos(2*x)+ cosh(2*y))	Tan[x + y*I] == Divide[Sin[2*x]+ I*Sinh[2*y],Cos[2*x]+ Cosh[2*y]]	Successful	Successful	-	Successful [Tested: 18]
4.21.E40	${\displaystyle{\displaystyle\cot z=\frac{\sin\left(2x\right)-\mathrm{i}\sinh\left(2y\right)}{\cosh\left(2y\right)-\cos\left(2x\right)}}}$ \cot@@{z} = \frac{\sin@{2x}-\iunit\sinh@{2y}}{\cosh@{2y}-\cos@{2x}}		cot(x + y*I) = (sin(2*x)- I*sinh(2*y))/(cosh(2*y)- cos(2*x))	Cot[x + y*I] == Divide[Sin[2*x]- I*Sinh[2*y],Cosh[2*y]- Cos[2*x]]	Successful	Successful	-	Successful [Tested: 18]
4.21.E41	${\displaystyle{\displaystyle|\sin z|=({\sin^{2}}x+{\sinh^{2}}y)^{1/2}}}$ |\sin@@{z}| = (\sin^{2}@@{x}+\sinh^{2}@@{y})^{1/2}		abs(sin(x + y*I)) = ((sin(x))^(2)+ (sinh(y))^(2))^(1/2)	Abs[Sin[x + y*I]] == ((Sin[x])^(2)+ (Sinh[y])^(2))^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.21.E41	${\displaystyle{\displaystyle({\sin^{2}}x+{\sinh^{2}}y)^{1/2}=\left(\tfrac{1}{2}\left(\cosh\left(2y\right)-\cos\left(2x\right)\right)\right)^{1/2}}}$ (\sin^{2}@@{x}+\sinh^{2}@@{y})^{1/2} = \left(\tfrac{1}{2}\left(\cosh@{2y}-\cos@{2x}\right)\right)^{1/2}		((sin(x))^(2)+ (sinh(y))^(2))^(1/2) = ((1)/(2)*(cosh(2*y)- cos(2*x)))^(1/2)	((Sin[x])^(2)+ (Sinh[y])^(2))^(1/2) == (Divide[1,2]*(Cosh[2*y]- Cos[2*x]))^(1/2)	Successful	Successful	-	Successful [Tested: 18]
4.21.E42	${\displaystyle{\displaystyle|\cos z|=({\cos^{2}}x+{\sinh^{2}}y)^{1/2}}}$ |\cos@@{z}| = (\cos^{2}@@{x}+\sinh^{2}@@{y})^{1/2}		abs(cos(x + y*I)) = ((cos(x))^(2)+ (sinh(y))^(2))^(1/2)	Abs[Cos[x + y*I]] == ((Cos[x])^(2)+ (Sinh[y])^(2))^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.21.E42	${\displaystyle{\displaystyle({\cos^{2}}x+{\sinh^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2y\right)+\cos\left(2x\right))\right)^{1/2}}}$ (\cos^{2}@@{x}+\sinh^{2}@@{y})^{1/2} = \left(\tfrac{1}{2}(\cosh@{2y}+\cos@{2x})\right)^{1/2}		((cos(x))^(2)+ (sinh(y))^(2))^(1/2) = ((1)/(2)*(cosh(2*y)+ cos(2*x)))^(1/2)	((Cos[x])^(2)+ (Sinh[y])^(2))^(1/2) == (Divide[1,2]*(Cosh[2*y]+ Cos[2*x]))^(1/2)	Successful	Successful	-	Successful [Tested: 18]
4.21.E43	${\displaystyle{\displaystyle|\tan z|=\left(\frac{\cosh\left(2y\right)-\cos\left(2x\right)}{\cosh\left(2y\right)+\cos\left(2x\right)}\right)^{1/2}}}$ |\tan@@{z}| = \left(\frac{\cosh@{2y}-\cos@{2x}}{\cosh@{2y}+\cos@{2x}}\right)^{1/2}		abs(tan(x + y*I)) = ((cosh(2*y)- cos(2*x))/(cosh(2*y)+ cos(2*x)))^(1/2)	Abs[Tan[x + y*I]] == (Divide[Cosh[2*y]- Cos[2*x],Cosh[2*y]+ Cos[2*x]])^(1/2)	Successful	Failure	-	Successful [Tested: 18]