First slide

Properties of ITF

Question

$\cot^{- 1} (\sqrt{\cos α}) - \tan^{- 1} (\sqrt{\cos α}) = x$ , then sin x =

Moderate

A

$\tan^{2} (\frac{α}{2})$
B

$\cot^{2} (\frac{α}{2})$
C

$\tan α$
D

$\cot (\frac{α}{2})$

Solution

$\begin{array}{l} \cot^{- 1} (\sqrt{\cos α}) - \tan^{- 1} (\sqrt{\cos α}) = x \\ \Rightarrow \tan^{- 1} (\frac{1}{\sqrt{\cos α}}) - \tan^{- 1} (\sqrt{\cos α}) = x \\ \Rightarrow \tan^{- 1} \frac{\frac{1}{\sqrt{\cos α}} - \sqrt{\cos α}}{1 + \frac{1}{\sqrt{\cos α}} \sqrt{\cos α}} = x \\ \Rightarrow \tan^{- 1} \frac{1 - \cos α}{2 \sqrt{\cos α}} = x \Rightarrow \tan x = \frac{1 - \cos α}{2 \sqrt{\cos α}} \\ or \cot x = \frac{2 \sqrt{\cos α}}{1 - \cos α} or cosec x = \frac{1 + \cos α}{1 - \cos α} \\ ∴ \sin x = \frac{1 - \cos α}{1 + \cos α} = \frac{1 - (1 - 2 \sin^{2} α / 2)}{1 + 2 \cos^{2} α / 2 - 1} \end{array}$
$or \sin x = \tan^{2} α / 2$

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