Solving $\;2^{\large \cos x} = \sin x$ $$2^{\large \cos x} = |\sin x|$$
Solve the equation. I found just one solution $\cos x= 0$ and are there any other solutions. Right hand side is modulus $\sin x$.

It would be enough to find the intersection of two graphs $2^{\cos x}$ and $\sin x$ over interval $[0,2\pi ]$ and since the graph again repeats after this the other solutions cvan be found by just adding $2k\pi$ where $k=0,1,2...$ to the intersection points. Hence plotting the graph once will help you a lot.
