Elementary Functions - 4.35 Identities

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.35.E1	${\displaystyle{\displaystyle\sinh\left(u+v\right)=\sinh u\cosh v+\cosh u\sinh v}}$ \sinh@{u+ v} = \sinh@@{u}\cosh@@{v}+\cosh@@{u}\sinh@@{v}		sinh(u + v) = sinh(u)*cosh(v)+ cosh(u)*sinh(v)	Sinh[u + v] == Sinh[u]*Cosh[v]+ Cosh[u]*Sinh[v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E1	${\displaystyle{\displaystyle\sinh\left(u-v\right)=\sinh u\cosh v-\cosh u\sinh v}}$ \sinh@{u- v} = \sinh@@{u}\cosh@@{v}-\cosh@@{u}\sinh@@{v}		sinh(u - v) = sinh(u)*cosh(v)- cosh(u)*sinh(v)	Sinh[u - v] == Sinh[u]*Cosh[v]- Cosh[u]*Sinh[v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E2	${\displaystyle{\displaystyle\cosh\left(u+v\right)=\cosh u\cosh v+\sinh u\sinh v}}$ \cosh@{u+ v} = \cosh@@{u}\cosh@@{v}+\sinh@@{u}\sinh@@{v}		cosh(u + v) = cosh(u)*cosh(v)+ sinh(u)*sinh(v)	Cosh[u + v] == Cosh[u]*Cosh[v]+ Sinh[u]*Sinh[v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E2	${\displaystyle{\displaystyle\cosh\left(u-v\right)=\cosh u\cosh v-\sinh u\sinh v}}$ \cosh@{u- v} = \cosh@@{u}\cosh@@{v}-\sinh@@{u}\sinh@@{v}		cosh(u - v) = cosh(u)*cosh(v)- sinh(u)*sinh(v)	Cosh[u - v] == Cosh[u]*Cosh[v]- Sinh[u]*Sinh[v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E3	${\displaystyle{\displaystyle\tanh\left(u+v\right)=\frac{\tanh u+\tanh v}{1+\tanh u\tanh v}}}$ \tanh@{u+ v} = \frac{\tanh@@{u}+\tanh@@{v}}{1+\tanh@@{u}\tanh@@{v}}		tanh(u + v) = (tanh(u)+ tanh(v))/(1 + tanh(u)*tanh(v))	Tanh[u + v] == Divide[Tanh[u]+ Tanh[v],1 + Tanh[u]*Tanh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E3	${\displaystyle{\displaystyle\tanh\left(u-v\right)=\frac{\tanh u-\tanh v}{1-\tanh u\tanh v}}}$ \tanh@{u- v} = \frac{\tanh@@{u}-\tanh@@{v}}{1-\tanh@@{u}\tanh@@{v}}		tanh(u - v) = (tanh(u)- tanh(v))/(1 - tanh(u)*tanh(v))	Tanh[u - v] == Divide[Tanh[u]- Tanh[v],1 - Tanh[u]*Tanh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E4	${\displaystyle{\displaystyle\coth\left(u+v\right)=\frac{+\coth u\coth v+1}{\coth u+\coth v}}}$ \coth@{u+ v} = \frac{+\coth@@{u}\coth@@{v}+1}{\coth@@{u}+\coth@@{v}}		coth(u + v) = (+ coth(u)*coth(v)+ 1)/(coth(u)+ coth(v))	Coth[u + v] == Divide[+ Coth[u]*Coth[v]+ 1,Coth[u]+ Coth[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[4.333014420201075*^14, -2.3525621062227262*^14] 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
4.35.E4	${\displaystyle{\displaystyle\coth\left(u-v\right)=\frac{-\coth u\coth v+1}{\coth u-\coth v}}}$ \coth@{u- v} = \frac{-\coth@@{u}\coth@@{v}+1}{\coth@@{u}-\coth@@{v}}		coth(u - v) = (- coth(u)*coth(v)+ 1)/(coth(u)- coth(v))	Coth[u - v] == Divide[- Coth[u]*Coth[v]+ 1,Coth[u]- Coth[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
4.35.E5	${\displaystyle{\displaystyle\sinh u+\sinh v=2\sinh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right)}}$ \sinh@@{u}+\sinh@@{v} = 2\sinh@{\frac{u+v}{2}}\cosh@{\frac{u-v}{2}}		sinh(u)+ sinh(v) = 2*sinh((u + v)/(2))*cosh((u - v)/(2))	Sinh[u]+ Sinh[v] == 2*Sinh[Divide[u + v,2]]*Cosh[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E6	${\displaystyle{\displaystyle\sinh u-\sinh v=2\cosh\left(\frac{u+v}{2}\right)\sinh\left(\frac{u-v}{2}\right)}}$ \sinh@@{u}-\sinh@@{v} = 2\cosh@{\frac{u+v}{2}}\sinh@{\frac{u-v}{2}}		sinh(u)- sinh(v) = 2*cosh((u + v)/(2))*sinh((u - v)/(2))	Sinh[u]- Sinh[v] == 2*Cosh[Divide[u + v,2]]*Sinh[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E7	${\displaystyle{\displaystyle\cosh u+\cosh v=2\cosh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right)}}$ \cosh@@{u}+\cosh@@{v} = 2\cosh@{\frac{u+v}{2}}\cosh@{\frac{u-v}{2}}		cosh(u)+ cosh(v) = 2*cosh((u + v)/(2))*cosh((u - v)/(2))	Cosh[u]+ Cosh[v] == 2*Cosh[Divide[u + v,2]]*Cosh[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E8	${\displaystyle{\displaystyle\cosh u-\cosh v=2\sinh\left(\frac{u+v}{2}\right)\sinh\left(\frac{u-v}{2}\right)}}$ \cosh@@{u}-\cosh@@{v} = 2\sinh@{\frac{u+v}{2}}\sinh@{\frac{u-v}{2}}		cosh(u)- cosh(v) = 2*sinh((u + v)/(2))*sinh((u - v)/(2))	Cosh[u]- Cosh[v] == 2*Sinh[Divide[u + v,2]]*Sinh[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E9	${\displaystyle{\displaystyle\tanh u+\tanh v=\frac{\sinh\left(u+v\right)}{\cosh u\cosh v}}}$ \tanh@@{u}+\tanh@@{v} = \frac{\sinh@{u+ v}}{\cosh@@{u}\cosh@@{v}}		tanh(u)+ tanh(v) = (sinh(u + v))/(cosh(u)*cosh(v))	Tanh[u]+ Tanh[v] == Divide[Sinh[u + v],Cosh[u]*Cosh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E9	${\displaystyle{\displaystyle\tanh u-\tanh v=\frac{\sinh\left(u-v\right)}{\cosh u\cosh v}}}$ \tanh@@{u}-\tanh@@{v} = \frac{\sinh@{u- v}}{\cosh@@{u}\cosh@@{v}}		tanh(u)- tanh(v) = (sinh(u - v))/(cosh(u)*cosh(v))	Tanh[u]- Tanh[v] == Divide[Sinh[u - v],Cosh[u]*Cosh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E10	${\displaystyle{\displaystyle\coth u+\coth v=\frac{\sinh\left(v+u\right)}{\sinh u\sinh v}}}$ \coth@@{u}+\coth@@{v} = \frac{\sinh@{v+ u}}{\sinh@@{u}\sinh@@{v}}		coth(u)+ coth(v) = (sinh(v + u))/(sinh(u)*sinh(v))	Coth[u]+ Coth[v] == Divide[Sinh[v + u],Sinh[u]*Sinh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E10	${\displaystyle{\displaystyle\coth u-\coth v=\frac{\sinh\left(v-u\right)}{\sinh u\sinh v}}}$ \coth@@{u}-\coth@@{v} = \frac{\sinh@{v- u}}{\sinh@@{u}\sinh@@{v}}		coth(u)- coth(v) = (sinh(v - u))/(sinh(u)*sinh(v))	Coth[u]- Coth[v] == Divide[Sinh[v - u],Sinh[u]*Sinh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E11	${\displaystyle{\displaystyle{\cosh^{2}}z-{\sinh^{2}}z=1}}$ \cosh^{2}@@{z}-\sinh^{2}@@{z} = 1		(cosh(z))^(2)- (sinh(z))^(2) = 1	(Cosh[z])^(2)- (Sinh[z])^(2) == 1	Successful	Successful	-	Successful [Tested: 7]
4.35.E12	${\displaystyle{\displaystyle{\operatorname{sech}^{2}}z=1-{\tanh^{2}}z}}$ \sech^{2}@@{z} = 1-\tanh^{2}@@{z}		(sech(z))^(2) = 1 - (tanh(z))^(2)	(Sech[z])^(2) == 1 - (Tanh[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.35.E13	${\displaystyle{\displaystyle{\operatorname{csch}^{2}}z={\coth^{2}}z-1}}$ \csch^{2}@@{z} = \coth^{2}@@{z}-1		(csch(z))^(2) = (coth(z))^(2)- 1	(Csch[z])^(2) == (Coth[z])^(2)- 1	Successful	Successful	-	Successful [Tested: 7]
4.35.E14	${\displaystyle{\displaystyle 2\sinh u\sinh v=\cosh\left(u+v\right)-\cosh\left(u-v\right)}}$ 2\sinh@@{u}\sinh@@{v} = \cosh@{u+v}-\cosh@{u-v}		2*sinh(u)*sinh(v) = cosh(u + v)- cosh(u - v)	2*Sinh[u]*Sinh[v] == Cosh[u + v]- Cosh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E15	${\displaystyle{\displaystyle 2\cosh u\cosh v=\cosh\left(u+v\right)+\cosh\left(u-v\right)}}$ 2\cosh@@{u}\cosh@@{v} = \cosh@{u+v}+\cosh@{u-v}		2*cosh(u)*cosh(v) = cosh(u + v)+ cosh(u - v)	2*Cosh[u]*Cosh[v] == Cosh[u + v]+ Cosh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E16	${\displaystyle{\displaystyle 2\sinh u\cosh v=\sinh\left(u+v\right)+\sinh\left(u-v\right)}}$ 2\sinh@@{u}\cosh@@{v} = \sinh@{u+v}+\sinh@{u-v}		2*sinh(u)*cosh(v) = sinh(u + v)+ sinh(u - v)	2*Sinh[u]*Cosh[v] == Sinh[u + v]+ Sinh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E17	${\displaystyle{\displaystyle{\sinh^{2}}u-{\sinh^{2}}v=\sinh\left(u+v\right)\sinh\left(u-v\right)}}$ \sinh^{2}@@{u}-\sinh^{2}@@{v} = \sinh@{u+v}\sinh@{u-v}		(sinh(u))^(2)- (sinh(v))^(2) = sinh(u + v)*sinh(u - v)	(Sinh[u])^(2)- (Sinh[v])^(2) == Sinh[u + v]*Sinh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E18	${\displaystyle{\displaystyle{\cosh^{2}}u-{\cosh^{2}}v=\sinh\left(u+v\right)\sinh\left(u-v\right)}}$ \cosh^{2}@@{u}-\cosh^{2}@@{v} = \sinh@{u+v}\sinh@{u-v}		(cosh(u))^(2)- (cosh(v))^(2) = sinh(u + v)*sinh(u - v)	(Cosh[u])^(2)- (Cosh[v])^(2) == Sinh[u + v]*Sinh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E19	${\displaystyle{\displaystyle{\sinh^{2}}u+{\cosh^{2}}v=\cosh\left(u+v\right)\cosh\left(u-v\right)}}$ \sinh^{2}@@{u}+\cosh^{2}@@{v} = \cosh@{u+v}\cosh@{u-v}		(sinh(u))^(2)+ (cosh(v))^(2) = cosh(u + v)*cosh(u - v)	(Sinh[u])^(2)+ (Cosh[v])^(2) == Cosh[u + v]*Cosh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E20	${\displaystyle{\displaystyle\sinh\frac{z}{2}=\left(\frac{\cosh z-1}{2}\right)^{1/2}}}$ \sinh@@{\frac{z}{2}} = \left(\frac{\cosh@@{z}-1}{2}\right)^{1/2}		sinh((z)/(2)) = ((cosh(z)- 1)/(2))^(1/2)	Sinh[Divide[z,2]] == (Divide[Cosh[z]- 1,2])^(1/2)	Failure	Failure	Failed [2 / 7] Result: -.4585952894+.8655770340*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: -.8655716642-.5419255224*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[-0.4585952893468803, 0.8655770337160631] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8655716640572735, -0.5419255224573363] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.35.E21	${\displaystyle{\displaystyle\cosh\frac{z}{2}=\left(\frac{\cosh z+1}{2}\right)^{1/2}}}$ \cosh@@{\frac{z}{2}} = \left(\frac{\cosh@@{z}+1}{2}\right)^{1/2}		cosh((z)/(2)) = ((cosh(z)+ 1)/(2))^(1/2)	Cosh[Divide[z,2]] == (Divide[Cosh[z]+ 1,2])^(1/2)	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.35.E22	${\displaystyle{\displaystyle\tanh\frac{z}{2}=\left(\frac{\cosh z-1}{\cosh z+1}\right)^{1/2}}}$ \tanh@@{\frac{z}{2}} = \left(\frac{\cosh@@{z}-1}{\cosh@@{z}+1}\right)^{1/2}		tanh((z)/(2)) = ((cosh(z)- 1)/(cosh(z)+ 1))^(1/2)	Tanh[Divide[z,2]] == (Divide[Cosh[z]- 1,Cosh[z]+ 1])^(1/2)	Failure	Failure	Failed [2 / 7] Result: -.5869891489+.8580864930*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: -.8595320616-.4211742148*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[-0.5869891488727425, 0.858086492859854] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8595320613685857, -0.42117421488499707] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.35.E22	${\displaystyle{\displaystyle\left(\frac{\cosh z-1}{\cosh z+1}\right)^{1/2}=\frac{\cosh z-1}{\sinh z}}}$ \left(\frac{\cosh@@{z}-1}{\cosh@@{z}+1}\right)^{1/2} = \frac{\cosh@@{z}-1}{\sinh@@{z}}		((cosh(z)- 1)/(cosh(z)+ 1))^(1/2) = (cosh(z)- 1)/(sinh(z))	(Divide[Cosh[z]- 1,Cosh[z]+ 1])^(1/2) == Divide[Cosh[z]- 1,Sinh[z]]	Failure	Failure	Failed [2 / 7] Result: .5869891489-.8580864930*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: .8595320615+.4211742148*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[0.5869891488727426, -0.8580864928598539] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.859532061368586, 0.42117421488499684] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.35.E22	${\displaystyle{\displaystyle\frac{\cosh z-1}{\sinh z}=\frac{\sinh z}{\cosh z+1}}}$ \frac{\cosh@@{z}-1}{\sinh@@{z}} = \frac{\sinh@@{z}}{\cosh@@{z}+1}		(cosh(z)- 1)/(sinh(z)) = (sinh(z))/(cosh(z)+ 1)	Divide[Cosh[z]- 1,Sinh[z]] == Divide[Sinh[z],Cosh[z]+ 1]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
4.35.E23	${\displaystyle{\displaystyle\sinh\left(-z\right)=-\sinh z}}$ \sinh@{-z} = -\sinh@@{z}		sinh(- z) = - sinh(z)	Sinh[- z] == - Sinh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E24	${\displaystyle{\displaystyle\cosh\left(-z\right)=\cosh z}}$ \cosh@{-z} = \cosh@@{z}		cosh(- z) = cosh(z)	Cosh[- z] == Cosh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E25	${\displaystyle{\displaystyle\tanh\left(-z\right)=-\tanh z}}$ \tanh@{-z} = -\tanh@@{z}		tanh(- z) = - tanh(z)	Tanh[- z] == - Tanh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E26	${\displaystyle{\displaystyle\sinh\left(2z\right)=2\sinh z\cosh z}}$ \sinh@{2z} = 2\sinh@@{z}\cosh@@{z}		sinh(2*z) = 2*sinh(z)*cosh(z)	Sinh[2*z] == 2*Sinh[z]*Cosh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E26	${\displaystyle{\displaystyle 2\sinh z\cosh z=\frac{2\tanh z}{1-{\tanh^{2}}z}}}$ 2\sinh@@{z}\cosh@@{z} = \frac{2\tanh@@{z}}{1-\tanh^{2}@@{z}}		2*sinh(z)*cosh(z) = (2*tanh(z))/(1 - (tanh(z))^(2))	2*Sinh[z]*Cosh[z] == Divide[2*Tanh[z],1 - (Tanh[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.35.E27	${\displaystyle{\displaystyle\cosh\left(2z\right)=2{\cosh^{2}}z-1}}$ \cosh@{2z} = 2\cosh^{2}@@{z}-1		cosh(2*z) = 2*(cosh(z))^(2)- 1	Cosh[2*z] == 2*(Cosh[z])^(2)- 1	Successful	Successful	-	Successful [Tested: 7]
4.35.E27	${\displaystyle{\displaystyle 2{\cosh^{2}}z-1=2{\sinh^{2}}z+1\\ }}$ 2\cosh^{2}@@{z}-1 = 2\sinh^{2}@@{z}+1\\		2*(cosh(z))^(2)- 1 = 2*(sinh(z))^(2)+ 1	2*(Cosh[z])^(2)- 1 == 2*(Sinh[z])^(2)+ 1	Successful	Successful	-	Successful [Tested: 7]
4.35.E27	${\displaystyle{\displaystyle 2{\sinh^{2}}z+1\\ ={\cosh^{2}}z+{\sinh^{2}}z}}$ 2\sinh^{2}@@{z}+1\\ = \cosh^{2}@@{z}+\sinh^{2}@@{z}		2*(sinh(z))^(2)+ 1 = (cosh(z))^(2)+ (sinh(z))^(2)	2*(Sinh[z])^(2)+ 1 == (Cosh[z])^(2)+ (Sinh[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.35.E28	${\displaystyle{\displaystyle\tanh\left(2z\right)=\frac{2\tanh z}{1+{\tanh^{2}}z}}}$ \tanh@{2z} = \frac{2\tanh@@{z}}{1+\tanh^{2}@@{z}}		tanh(2*z) = (2*tanh(z))/(1 + (tanh(z))^(2))	Tanh[2*z] == Divide[2*Tanh[z],1 + (Tanh[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.35.E29	${\displaystyle{\displaystyle\sinh\left(3z\right)=3\sinh z+4{\sinh^{3}}z}}$ \sinh@{3z} = 3\sinh@@{z}+4\sinh^{3}@@{z}		sinh(3*z) = 3*sinh(z)+ 4*(sinh(z))^(3)	Sinh[3*z] == 3*Sinh[z]+ 4*(Sinh[z])^(3)	Successful	Successful	-	Successful [Tested: 7]
4.35.E30	${\displaystyle{\displaystyle\cosh\left(3z\right)=-3\cosh z+4{\cosh^{3}}z}}$ \cosh@{3z} = -3\cosh@@{z}+4\cosh^{3}@@{z}		cosh(3*z) = - 3*cosh(z)+ 4*(cosh(z))^(3)	Cosh[3*z] == - 3*Cosh[z]+ 4*(Cosh[z])^(3)	Successful	Successful	-	Successful [Tested: 7]
4.35.E31	${\displaystyle{\displaystyle\sinh\left(4z\right)=4{\sinh^{3}}z\cosh z+4{\cosh^{3}}z\sinh z}}$ \sinh@{4z} = 4\sinh^{3}@@{z}\cosh@@{z}+4\cosh^{3}@@{z}\sinh@@{z}		sinh(4*z) = 4*(sinh(z))^(3)* cosh(z)+ 4*(cosh(z))^(3)* sinh(z)	Sinh[4*z] == 4*(Sinh[z])^(3)* Cosh[z]+ 4*(Cosh[z])^(3)* Sinh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E32	${\displaystyle{\displaystyle\cosh\left(4z\right)={\cosh^{4}}z+6{\sinh^{2}}z{\cosh^{2}}z+{\sinh^{4}}z}}$ \cosh@{4z} = \cosh^{4}@@{z}+6\sinh^{2}@@{z}\cosh^{2}@@{z}+\sinh^{4}@@{z}		cosh(4*z) = (cosh(z))^(4)+ 6*(sinh(z))^(2)* (cosh(z))^(2)+ (sinh(z))^(4)	Cosh[4*z] == (Cosh[z])^(4)+ 6*(Sinh[z])^(2)* (Cosh[z])^(2)+ (Sinh[z])^(4)	Successful	Successful	-	Successful [Tested: 7]
4.35.E33	${\displaystyle{\displaystyle\cosh\left(nz\right)+\sinh\left(nz\right)=(\cosh z+\sinh z)^{n}}}$ \cosh@{nz}+\sinh@{nz} = (\cosh@@{z}+\sinh@@{z})^{n}		cosh(n*z)+ sinh(n*z) = (cosh(z)+ sinh(z))^(n)	Cosh[n*z]+ Sinh[n*z] == (Cosh[z]+ Sinh[z])^(n)	Successful	Successful	-	Successful [Tested: 7]
4.35.E33	${\displaystyle{\displaystyle\cosh\left(nz\right)-\sinh\left(nz\right)=(\cosh z-\sinh z)^{n}}}$ \cosh@{nz}-\sinh@{nz} = (\cosh@@{z}-\sinh@@{z})^{n}		cosh(n*z)- sinh(n*z) = (cosh(z)- sinh(z))^(n)	Cosh[n*z]- Sinh[n*z] == (Cosh[z]- Sinh[z])^(n)	Successful	Successful	-	Successful [Tested: 7]
4.35.E34	${\displaystyle{\displaystyle\sinh z=\sinh x\cos y+i\cosh x\sin y}}$ \sinh@@{z} = \sinh@@{x}\cos@@{y}+i\cosh@@{x}\sin@@{y}		sinh(x + y*I) = sinh(x)*cos(y)+ I*cosh(x)*sin(y)	Sinh[x + y*I] == Sinh[x]*Cos[y]+ I*Cosh[x]*Sin[y]	Successful	Successful	-	Successful [Tested: 18]
4.35.E35	${\displaystyle{\displaystyle\cosh z=\cosh x\cos y+i\sinh x\sin y}}$ \cosh@@{z} = \cosh@@{x}\cos@@{y}+i\sinh@@{x}\sin@@{y}		cosh(x + y*I) = cosh(x)*cos(y)+ I*sinh(x)*sin(y)	Cosh[x + y*I] == Cosh[x]*Cos[y]+ I*Sinh[x]*Sin[y]	Successful	Successful	-	Successful [Tested: 18]
4.35.E36	${\displaystyle{\displaystyle\tanh z=\frac{\sinh\left(2x\right)+i\sin\left(2y\right)}{\cosh\left(2x\right)+\cos\left(2y\right)}}}$ \tanh@@{z} = \frac{\sinh@{2x}+i\sin@{2y}}{\cosh@{2x}+\cos@{2y}}		tanh(x + y*I) = (sinh(2*x)+ I*sin(2*y))/(cosh(2*x)+ cos(2*y))	Tanh[x + y*I] == Divide[Sinh[2*x]+ I*Sin[2*y],Cosh[2*x]+ Cos[2*y]]	Successful	Successful	-	Successful [Tested: 18]
4.35.E37	${\displaystyle{\displaystyle\coth z=\frac{\sinh\left(2x\right)-i\sin\left(2y\right)}{\cosh\left(2x\right)-\cos\left(2y\right)}}}$ \coth@@{z} = \frac{\sinh@{2x}-i\sin@{2y}}{\cosh@{2x}-\cos@{2y}}		coth(x + y*I) = (sinh(2*x)- I*sin(2*y))/(cosh(2*x)- cos(2*y))	Coth[x + y*I] == Divide[Sinh[2*x]- I*Sin[2*y],Cosh[2*x]- Cos[2*y]]	Successful	Successful	-	Successful [Tested: 18]
4.35.E38	${\displaystyle{\displaystyle|\sinh z|=({\sinh^{2}}x+{\sin^{2}}y)^{1/2}}}$ |\sinh@@{z}| = (\sinh^{2}@@{x}+\sin^{2}@@{y})^{1/2}		abs(sinh(x + y*I)) = ((sinh(x))^(2)+ (sin(y))^(2))^(1/2)	Abs[Sinh[x + y*I]] == ((Sinh[x])^(2)+ (Sin[y])^(2))^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.35.E38	${\displaystyle{\displaystyle({\sinh^{2}}x+{\sin^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2x\right)-\cos\left(2y\right))\right)^{1/2}}}$ (\sinh^{2}@@{x}+\sin^{2}@@{y})^{1/2} = \left(\tfrac{1}{2}(\cosh@{2x}-\cos@{2y})\right)^{1/2}		((sinh(x))^(2)+ (sin(y))^(2))^(1/2) = ((1)/(2)*(cosh(2*x)- cos(2*y)))^(1/2)	((Sinh[x])^(2)+ (Sin[y])^(2))^(1/2) == (Divide[1,2]*(Cosh[2*x]- Cos[2*y]))^(1/2)	Successful	Successful	-	Successful [Tested: 18]
4.35.E39	${\displaystyle{\displaystyle|\cosh z|=({\sinh^{2}}x+{\cos^{2}}y)^{1/2}}}$ |\cosh@@{z}| = (\sinh^{2}@@{x}+\cos^{2}@@{y})^{1/2}		abs(cosh(x + y*I)) = ((sinh(x))^(2)+ (cos(y))^(2))^(1/2)	Abs[Cosh[x + y*I]] == ((Sinh[x])^(2)+ (Cos[y])^(2))^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.35.E39	${\displaystyle{\displaystyle({\sinh^{2}}x+{\cos^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2x\right)+\cos\left(2y\right))\right)^{1/2}}}$ (\sinh^{2}@@{x}+\cos^{2}@@{y})^{1/2} = \left(\tfrac{1}{2}(\cosh@{2x}+\cos@{2y})\right)^{1/2}		((sinh(x))^(2)+ (cos(y))^(2))^(1/2) = ((1)/(2)*(cosh(2*x)+ cos(2*y)))^(1/2)	((Sinh[x])^(2)+ (Cos[y])^(2))^(1/2) == (Divide[1,2]*(Cosh[2*x]+ Cos[2*y]))^(1/2)	Successful	Successful	-	Successful [Tested: 18]
4.35.E40	${\displaystyle{\displaystyle|\tanh z|=\left(\frac{\cosh\left(2x\right)-\cos\left(2y\right)}{\cosh\left(2x\right)+\cos\left(2y\right)}\right)^{1/2}}}$ |\tanh@@{z}| = \left(\frac{\cosh@{2x}-\cos@{2y}}{\cosh@{2x}+\cos@{2y}}\right)^{1/2}		abs(tanh(x + y*I)) = ((cosh(2*x)- cos(2*y))/(cosh(2*x)+ cos(2*y)))^(1/2)	Abs[Tanh[x + y*I]] == (Divide[Cosh[2*x]- Cos[2*y],Cosh[2*x]+ Cos[2*y]])^(1/2)	Successful	Failure	-	Successful [Tested: 18]