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]

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]

We have, ${\tan ^{ – 1}}\left( { – \frac{1}{{\sqrt 3 }}} \right) + {\cot ^{ – 1}}\left( {\frac{1}{{\sqrt 3 }}} \right) + {\tan ^{ – 1}}\left[ {\sin \left( {\frac{{ – \pi }}{2}} \right)} \right]$

$ = {\tan ^{ – 1}}\left( {\tan \frac{{5\pi }}{6}} \right) + {\cot ^{ – 1}}\left( {\cot \frac{\pi }{3}} \right) + {\tan ^{ – 1}}( – 1)$

$ = {\tan ^{ – 1}}\left[ {\tan \left( {\pi – \frac{\pi }{6}} \right)} \right] + {\cot ^{ – 1}}\left[ {\cot \left( {\frac{\pi }{3}} \right)} \right] + {\tan ^{ – 1}}\left[ {\tan \left( {\pi – \frac{\pi }{4}} \right)} \right]$

$ = {\tan ^{ – 1}}\left( { – \tan \frac{\pi }{6}} \right) + {\cot ^{ – 1}}\left( {\cot \frac{\pi }{3}} \right) + {\tan ^{ – 1}}\left( { – \tan \frac{\pi }{4}} \right)$

$ = – \frac{\pi }{6} + \frac{\pi }{3} – \frac{\pi }{4} = \frac{{ – 2\pi + 4\pi – 3\pi }}{{12}}$
$ = \frac{{ – 5\pi + 4\pi }}{{12}} = – \frac{\pi }{{12}}$

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