Question3. Prove the following identities: (a) $\sinh (z+\pi)=-\sinh (z)$ (b) $\cosh (z)=\cosh (x) \cos (y)+i \sinh (x) \sin (y)$

Studdy Solution
Recognize the hyperbolic identities:
$e^x + e^{-x} = 2\cosh(x) \quad \text{and} \quad e^x - e^{-x} = 2\sinh(x)$
Substitute these into the expression:
$\cosh(z) = \cosh(x)\cos(y) + i\sinh(x)\sin(y)$
The identities have been proven:
(a) $\sinh(z+\pi) = -\sinh(z)$
(b) $\cosh(z) = \cosh(x) \cos(y) + i \sinh(x) \sin(y)$