Question 106
Find all real numbers $a$ and $b$ so that for all real numbers $x$,
$$
2 \cos^2\left(x+\frac{b}{2}\right) - 2 \sin\left(a x - \frac{\pi}{2}\right) \cos\left(a x - \frac{\pi}{2}\right) = 1
$$
$$
2 \cos^2\left(x+\frac{b}{2}\right) - 2 \sin\left(a x - \frac{\pi}{2}\right) \cos\left(a x - \frac{\pi}{2}\right) = 1
$$