Here I have this trigonometric equation $\cos 7\theta=\cos 3\theta+\sin 5\theta$.
I approached the problem as follows : $\cos 7\theta=\cos 3\theta+\sin 5\theta$ $\implies \cos 7\theta-\cos 3\theta=\sin 5\theta$ $\implies 2\sin 5\theta\sin(-2\theta)=\sin 5\theta$
From there on we get, $\sin 2\theta=-\dfrac{1}{2}$ and $\sin 5\theta=0$
So, we deduce that $\theta=-\dfrac{\pi}{12}+\pi k$ and $\theta=\dfrac{2}{5}\pi n$ where $k$ and $n$ are integers.
However, Wolfram|Alpha doesn't seem to agree with me. They present $\dfrac{\pi}{5}$ as a solution which is not included in my solution.
Can someone help me out?