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.
[NCERT,Exemplar.2.3,Q.9,Page.36]

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}$

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