Hello!

I was asked to show that the following identity holds:

|cosh(z)|^{2}= sinh^{2}(x) + cos^{2}(y)

This is what I've got,

LS= |cosh(z)| ^{2 }= |1/2(e^{z}+e^{-}^{z})|^{2}

=1/4(e^{z}+e^{-}^{z})(e^{z}+e^{-}^{z})

=1/4(e^{2}^{z}+2+e^{-2}^{z})RS= sinh ^{2}(x) + cos^{2}(y)

= (1/2)(e^{x}-e^{-}^{x})(1/2)(e^{x}-e^{-}^{x})

+ (1/2)(e^{iy}+e^{-}^{iy})(1/2)(e^{iy}+e^{-}^{iy})

= 1/4(e^{2x}-2+e^{-2x}+e^{2iy}+2+e^{-2iy})

= 1/4(e^{2}^{z}+e^{-2}^{z})

So LS does not equal RS....

Where is my mistake? Is it in the Absolute value set on the LS? that's the only part that is not intuitive for me...

Any hints would be killler!!!