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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?


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up vote 4 down vote accepted

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 $$ \def\abs#1{\left|#1\right|} \abs{\sin i}^2 + \abs{\cos i}^2 = \frac{1+e^4}{2e^2} \ne 1. $$

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thanks martini. – Sara Tancredi Oct 9 '13 at 11:25
Dear Martini, I tried to evaluate this equation by substituting $z$ to $\iota$. I expanded $sin\iota$ and $cos\iota$ in their corresponding series but at the end it didn't give me the value that you got. Please explain your calculation. – Sara Tancredi Oct 12 '13 at 7:11

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