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

We have, ${\tan ^{ – 1}}\left( {\tan \frac{{2\pi }}{3}} \right) = {\tan ^{ – 1}}\tan \left( {\pi – \frac{\pi }{3}} \right)$

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

$ = – {\tan ^{ – 1}}\tan \frac{\pi }{3} = – \frac{\pi }{3}$

Note
Remember that, ${\tan ^{ – 1}}\left( {\tan \frac{{2\pi }}{3}} \right) \ne \frac{{2\pi }}{3}$

Since, ${\tan ^{ – 1}}(\tan x) = x$, if $x \in \left( { – \frac{\pi }{2},\frac{\pi }{2}} \right)$

and $\frac{{2\pi }}{3} \notin \left( {\frac{{ – \pi }}{2},\frac{\pi }{2}} \right)$

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