Note that $\sin a + \sin 3a = 2\sin 2a \cdot \cos a$. Likewise, $\cos a + \cos 3a = 2\cos a \cdot \cos 2a$. Combining, we find \[ \sin 2a (1+2\cos a) = \cos 2a (1+2\cos a). \]If $\cos a =-1/2$, we easily see the sets don't coincide. This forces $a= \pi/8+k\cdot \pi/2$, $k\in\mathbb{Z}$. These angles work, concluding the problem.