Answer
$\frac{k\pi}{6}$
Work Step by Step
$\cos 5\theta-\cos 7\theta=0$
Use the Sum-to-Product Formula $\cos x-\cos y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$:
$-2\sin\frac{5\theta+7\theta}{2}\sin\frac{5\theta-7\theta}{2}=0$
$\sin\frac{12\theta}{2}\sin\frac{-2\theta}{2}=0$
$\sin 6\theta\sin(-\theta)=0$
If $\sin 6\theta=0$, then $6\theta=k\pi$, and $\theta=\frac{k\pi}{6}$.
If $\sin(-\theta)=0$, then $-\theta=k\pi$, and $\theta=-k\pi$. However, note that this is already included in the above solution, $\theta=\frac{k\pi}{6}$.
