Let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for $x \in \mathbb{R}$. Then the number of real solutions of the equation $\sqrt{1+\cos (2 x)}=\sqrt{2} \tan ^{-1} \tan (x)$ in the set $\left(-\frac{3 \pi}{2}, \frac{\pi}{2}\right) \cup\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \cup\left(-\frac{\pi}{2}, \frac{3 \pi}{2}\right)$ is equal to -