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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]

Find the value of $\sin \left( {2{{\tan }^{ – 1}}\frac{1}{3}} \right) + \cos \left( {{{\tan }^{ – 1}}2\sqrt 2 } \right)$. [NCERT,Exemplar.2.3,Q.8,Page.36]

Find the real Solution of
${\tan ^{ – 1}}\sqrt {x(x + 1)} + {\sin ^{ – 1}}\sqrt {{x^2} + x + 1} = \frac{\pi }{2}$.
[NCERT,Exemplar.2.3,Q.7,Page.36]

Show that $2{\tan ^{ – 1}}( – 3) = \frac{{ – \pi }}{2} + {\tan ^{ – 1}}\left( {\frac{{ – 4}}{3}} \right)$.

[NCERT,Exemplar.2.3,Q.6,Page.35]

Find the value of ${\tan ^{ – 1}}\left( {\tan \frac{{2\pi }}{3}} \right)$.
[NCERT,Exemplar.2.3,Q.5,Page.35]

Find the value of ${\tan ^{ – 1}}\left( { – \frac{1}{{\sqrt 3 }}} \right) + {\cot ^{ – 1}}\left( {\frac{1}{{\sqrt 3 }}} \right) + {\tan ^{ – 1}}\left[ {\sin \left( {\frac{{ – \pi }}{2}} \right)} \right]$.
[NCERT,Exemplar.2.3,Q.4,Page.35]

Prove that $\cot \left( {\frac{\pi }{4} – 2{{\cot }^{ – 1}}3} \right) = 7$.
[NCERT,Exemplar.2.3,Q.3,Page.35]

Evaluate $\cos \left[ {{{\cos }^{ – 1}}\left( {\frac{{ – \sqrt 3 }}{2}} \right) + \frac{\pi }{6}} \right]$.
[NCERT,Exemplar.2.3,Q.2,Page.35]

Find the value of ${\tan ^{ – 1}}\left( {\tan \frac{{5\pi }}{6}} \right) + {\cos ^{ – 1}}\left( {\cos \frac{{13\pi }}{6}} \right)$.
[NCERT,Exemplar.2.3,Q.1,Page.35]

Find the value of ${\tan ^{ – 1}}\left( {\tan \frac{{5\pi }}{6}} \right) + {\cos ^{ – 1}}\left( {\cos \frac{{13\pi }}{6}} \right)$.
[NCERT,Exemplar.2.3,Q.1,Page.35]

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