Question

Trigonometric Identity for complex, I don't know where's my wrong, please help me
|Cos h(z)|^2=sin h^2(x)+cos^2(y) .Prove , Check attachment

Answer 1

at step 2

to find the magnitude squared of a complex number a+ i b
multiply by the complex conjugate :
\( (a+ib)(a-ib)= a^2 - i^2 b^2= a^2+b^2\)

so your 2nd line should read
\[ \cosh^2(x) \cos^2(y) - i^2 \sinh^2(x) \sin^2(y) \]
and your 3rd line should be
\[ \cosh^2(x) \cos^2(y) + \sinh^2(x) \sin^2(y) \]