Consider the function $f(z)=-sin^2(z)+2cos(z)-2$
I'm asked to assume that $f(z+a)=f(z)$ and then show that $a=2\pi k$, k being an integer.
How would I go about this?
I tried setting $a=\pi$ and hoping $\pi$ would be an antiperiod of $f$ and thereby showing the period to be $2\pi k$ , but thats not the case. I feel lost!
There's a way of rewriting the problem that may make it simpler in your mind. Note that $\sin{z}=\frac{e^{iz}-e^{-iz}}{2i}$ and cosine is the same but with a factor of one half instead of 1 over i and two plus signs instead of one plus and one minus. Then write f in terms of these exponentials and collect them. You should start to see where you should seek periodicity (in the periodic exponential function). Can you take it from here?