Question

\[\text{ If } u = \cot^{- 1} \sqrt{\tan \theta} - \tan^{- 1} \sqrt{\tan \theta}\text{ then }, \tan\left( \frac{\pi}{4} - \frac{u}{2} \right) =\]

Options

• `sqrt(tantheta`

• `sqrt(cottheta)`

•  tan θ

• cot θ

Answer 1

(a) `sqrt(tantheta`
Let \[y = \sqrt{\tan\theta}\]
Then, 
\[u = \cot^{- 1} \sqrt{\tan\theta} - \tan^{- 1} \sqrt{\tan\theta}\]
\[ \Rightarrow u = \cot^{- 1} y - \tan^{- 1} y\]
\[ \Rightarrow u = \frac{\pi}{2} - 2 \tan^{- 1} y \left[ \because \tan^{- 1} x + \cot^{- 1} x = \frac{\pi}{2} \right]\]
\[ \Rightarrow 2 \tan^{- 1} y = \frac{\pi}{2} - u \]
\[ \Rightarrow \tan^{- 1} y = \frac{\pi}{4} - \frac{u}{2}\]
\[ \Rightarrow y = \tan\left( \frac{\pi}{4} - \frac{u}{2} \right)\]
\[ \Rightarrow \sqrt{\tan\theta} = \tan\left( \frac{\pi}{4} - \frac{u}{2} \right) \left[ \because y = \sqrt{\tan\theta} \right]\]
\[\]