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 $ |sinz|^2+|cosz|^2$ from $\Bbb R$ to $\Bbb C$? (i.e Can we say that $ |sinz|^2+|cosz|^2=1$$\forall z \in\Bbb C$)?
I think this extension of domain from real line to entire complex plane is not possible. Since function on left hand side i.e.$ |sinz|^2+|cosz|^2$ doesn't look analytic. And to use above theorem we need both functions to be analytic. How will we show this?