# Finding the largest set where a complex function is analytic

I am considering the function $$f(z) = \frac{e^z}{\sin z - \cos z}.$$

So I solved for $\sin z - \cos z = 0$ and got $\pi/4$. But why is it $\pi/4 + k\pi$ and not $\pi/4 + k2\pi$ for the part of the complex plane where this function is not analytic?

Thanks.

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$$\sin z -\cos z=0\Longleftrightarrow \tan z = 1\Longrightarrow z=\frac{\pi}{4}+k\pi\,\,,\,k\in\Bbb Z$$
as the period of the tangent function is $\,\pi\,$ , not $\,2\pi\,$