Question: WolframAlpha gives the following expression $$\sum_{k=1}^{n}\frac{\cos(\pi k)}{\csc(\pi k)}=\phi(n);$$ where $\phi()$ is the Euler totient function. You can check for yourself here. Is this incorrect ?
Surely $\frac{\cos(\pi k)}{\csc(\pi x)}=0$ for every natural number $k,$ and so the sum should always equal zero. Am I misinterpreting the output from WolframAlpha or did I type it in wrong ? I know that $\frac{\cos(\pi k)}{\csc(\pi x)}=\frac{1}{2}\sin(2\pi k),$ whose partial sum WolframAlpha correctly shows: check.