From $1=\sum_{k\geq 1} a_k \sin((k\pi+\frac{\pi}{2})x),$ I want to find $a_k.$
My unsuccessful approach is first multiplying both side by $\cos((k\pi+\frac{\pi}{2})x)$. That is, $\cos((k\pi+\frac{\pi}{2})x)=\frac{1}{2}\sum_{k\geq 1}(\sin \pi x)a_k.$
Can anyone give me a trick to find $a_k$? Thanks in advance.