Question

Solve for x : \[\cos \left( \tan^{- 1} x \right) = \sin \left( \cot^{- 1} \frac{3}{4} \right)\] .

Answer 1

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 

\[\cos\left( \tan^{- 1} x \right)\]  in equation (1), we get:

\[\sin\left( \cot^{- 1} x \right) = \sin\left( \cot^{- 1} \frac{3}{4} \right)\]

\[ \Rightarrow x = \frac{3}{4}\]