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