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Question
If \[\tan^{- 1} (\sqrt{3}) + \cot^{- 1} x = \frac{\pi}{2},\] find x.
Solution
We know that
\[\tan^{- 1} x + \cot^{- 1} x = \frac{\pi}{2}\]
We have
\[\tan^{- 1} \left( \sqrt{3} \right) + \cot^{- 1} x = \frac{\pi}{2}\]
\[ \Rightarrow \tan^{- 1} \left( \sqrt{3} \right) = \frac{\pi}{2} - \cot^{- 1} x\]
\[ \Rightarrow \tan^{- 1} \left( \sqrt{3} \right) = \tan^{- 1} x\]
\[ \Rightarrow x = \sqrt{3}\]
∴ \[x = \sqrt{3}\]
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