TITLE: Is it true that $ |\sin^2z+\cos^2z|=1, \forall z \in\Bbb C$?
QUESTION [6 upvotes]: We know that equation $ \sin^2z+ \cos^2z=1$ which holds  $ \forall z \in\Bbb R$, actually holds $ \forall z \in\Bbb C$.
Is it true that $ |\sin^2z+\cos^2z|=1, \forall z \in\Bbb C$?
Thanks in advance.

REPLY [11 votes]: In general, if an identity holds for a smooth function over real numbers, then the same identity also holds over the complex numbers. (Amazing!)
In this case: $$\cos^2(z)+\sin^2(z)=(\cos(z)+i\sin(z))(\cos(z)-i\sin(z))=e^{iz}e^{-iz}=1$$
where you need the exponential definition of trigonometric functions. There is no need for the absolute value in your question.