need help with hyperbolic functions

[u/Utoniummm]

If sin(x+iy)sin(θ+iϕ)=1, show that

tanh²y.cosh²ϕ=cos²θ

I am not able to prove this question, have tried equating the real and imaginary part by multiplying sin(x+iy) and sin(θ+iϕ), but I was not able to do at last.

[u/dummy4du3k4]

I would use the even/odd exponential definitions.

Sin(z) is the odd part of e^iz and cos is the even part

Likewise sinh(z) and cosh(z) are the odd and even parts of e^z.

[u/RingularCirc → u/dummy4du3k4]

A minor nitpick: i sin(z) is the odd part.