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Question
Solve for x : \[\cos \left( \tan^{- 1} x \right) = \sin \left( \cot^{- 1} \frac{3}{4} \right)\] .
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Solution
Given:
\[\cos \left( \tan^{- 1} x \right) = \sin \left( \cot^{- 1} \frac{3}{4} \right)\] .........(1)
\[cos\theta = \sin\left( \frac{\pi}{2} - \theta \right)\]
\[ \Rightarrow \cos\left( \tan^{- 1} x \right) = \sin\left( \frac{\pi}{2} - \tan^{- 1} x \right)\]
\[ \Rightarrow \cos\left( \tan^{- 1} x \right) = \sin\left( \cot^{- 1} x \right)\]
Substituting the value of
\[\sin\left( \cot^{- 1} x \right) = \sin\left( \cot^{- 1} \frac{3}{4} \right)\]
\[ \Rightarrow x = \frac{3}{4}\]
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