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Could anyone offer a proof that

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

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    $\begingroup$ Perhaps someone can offer a proof, but not a correct proof, because it isn't true. Consider the case $x=y=0$. $\endgroup$ – David Aug 4 '14 at 12:22
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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

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