Proof that $\cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)$

Question

Could anyone offer a proof that

$$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$

Answer 1

Note that $\sin^2 x+\cos^2 x=1$ and $\sinh^2 x=1+\cosh^2 x$

so the left hand side is

$$\cos^2 x \cosh^2 x+ \sin^2 x \sinh^2 x= (1-\sin^2 x) (1+\sinh^2 x)+ \sin^2 x \sinh^2 x= 1-\sin^2 x +\sinh^2 x $$

So the sign in the question is wrong