$ = {\tan ^{ - 1}}\left[ {\frac{{\sqrt {1 + \cos 2\theta } + \sqrt {1 - \cos 2\theta } }}{{\sqrt {1 + \cos 2\theta } - \sqrt {1 - \cos 2\theta } }}} \right]$
(Putting ${x^2} = \cos 2\theta \Rightarrow \theta = \frac{1}{2}{\cos ^{ - 1}}{x^2})$
$= {\tan ^{ - 1}}\left[ {\frac{{\sqrt 2 \cos \theta + \sqrt 2 \sin \theta }}{{\sqrt 2 \cos \theta - \sqrt 2 \sin \theta }}} \right]$
$ = {\tan ^{ - 1}}\left[ {\frac{{1 + \tan \theta }}{{1 - \tan \theta }}} \right] = {\tan ^{ - 1}}\left[ {\frac{{\tan \frac{\pi }{4} + \tan \theta }}{{1 - \tan \frac{\pi }{4}\tan \theta }}} \right]$
$ = {\tan ^{ - 1}}\tan \left( {\frac{\pi }{4} + \theta } \right) = \frac{\pi }{4} + \theta = \frac{\pi }{4} + \frac{1}{2}{\cos ^{ - 1}}{x^2}$.
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Statement $2$ : A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.