Prove that $\cos^2(z)+\sin^2(z)=1$ Can anybody show me how to prove that $\cos^2(z)+\sin^2(z)=1$ for $z\in \mathbb{C}$? I can prove it for the case where $z$ is real, but can't seem to find a way to prove it for complex numbers in general.

The very same proof that works for real values works for complex values: both $\sin z$ and $\cos z$ admit convergent powerseries on the whole complex plane, which you can differentiate term by term, giving that
$$(\cos z) ' = -\sin z$$
$$(\sin z)' = \cos z$$
If you consider $f(z) = \cos^2 z +\sin^2 z-1$ the above gives $f'(z) =0$:
$$f'(z) = -2\cos z\sin z+2\sin z\cos z=0$$ Thus $f$ must be constant, and because $f(0)=0$, it is constantly zero. 
