We know that the equation $ \sin^2z+ \cos^2z=1$ which holds $\forall z \in\Bbb R$, also holds $\forall z \in\Bbb C$.
This is obvious under the shadow of following theorem:
Uniqueness principle theorem :If $f$ and $g$ are analytic functions on a domain $D$, and if $f(z)=g(z)$ for $z$ belonging to a set that has a non isolated point, then $f(z)=g(z)$ for all $z\in D$.
Can we extend the domain of function $ |\sin z|^2+|\cos z|^2$ from $\Bbb R$ to $\Bbb C$? (i.e Can we say that $|\sin z|^2+|\cos z|^2=1\,\forall z \in\Bbb C$)?
I think this extension of the domain from the real line to the entire complex plane is not possible. Since function on left-hand side i.e.$ |\sin z|^2+|\cos z|^2$ doesn't look analytic. And to use above theorem we need both functions to be analytic. How will we show this?
Thanks.
As you say correctly, the function $z \mapsto |\sin z|^2 + |\cos z|^2$ is not analytic, so the comparision principle cannot be used. Moreover, want you want to prove is wrong, as - for example $$ \def\abs#1{\left|#1\right|} \abs{\sin i}^2 + \abs{\cos i}^2 = \frac{1+e^4}{2e^2} \ne 1. $$