If $2{\tan ^{ - 1}}(\cos \theta ) = {\tan ^{ - 1}}(2{\mathop{\rm cosec}\nolimits} \theta )$, then show that $\theta = \frac{\pi }{4}$, where $n$ is any integer.

We have, $2{\tan ^{ - 1}}(\cos \theta ) = {\tan ^{ - 1}}(2{\mathop{\rm cosec}\nolimits} \theta )$
$\Rightarrow$ ${\tan ^{ - 1}}\left( {\frac{{2\cos \theta }}{{1 - {{\cos }^2}\theta }}} \right) = {\tan ^{ - 1}}(2{\mathop{\rm cosec}\nolimits} \theta )$

$\Rightarrow \left( {\frac{{2\cos \theta }}{{{{\sin }^2}\theta }}} \right) = (2{\mathop{\rm cosec}\nolimits} \theta )$
$\Rightarrow (\cot \theta \cdot 2{\mathop{\rm cosec}\nolimits} \theta ) = (2{\mathop{\rm cosec}\nolimits} \theta ) \Rightarrow \cot \theta = 1$

$\Rightarrow$ $\cot \theta = \cot \frac{\pi }{4} \Rightarrow \theta = \frac{\pi }{4}$