Having trouble solving a problem involving hyperbolic trignometric functions

Question

We have to find the value of $$ \tanh^{2}a * \cosh^{2} b - \cos^ {2} c \, $$

if $$\sin(a+ib) * \sin(c+id) = 1.$$

Can anyone solve this? Pls share the solution

Answer 1

HINT:

$$\sin(a+ib)\cdot\sin(c+id)=1\implies\sin(a-ib)\cdot\sin(c-id)=1$$

$$\sin(a+ib)\cdot\sin(a-ib)=\sin^2a-\sin^2(ib)=\sin^2a-(i\sinh b)^2$$

and use $\cosh^2u-\sinh^2u=1$

Alternatively, $$\sin(a+ib)=\sin a\cos(ib)+\cos a\sin(ib)=\sin(a)\cosh(b)+i\cos(a)\sinh(b)$$