First slide

Properties of ITF

Question

The value of $\tan^{- 1} (\frac{1}{2} \tan 2 A) + \tan^{- 1} (\cot A) + \tan^{- 1} (\cot^{3} A)$ , for $0 < A < π / 4$ , is

Moderate

A

$\tan^{- 1} 2$
B

$\tan^{- 1} (\cot A)$
C

$4 \tan^{- 1} (1)$
D

$2 \tan^{- 1} (2)$

Solution

For $0 < A < π / 4$ , $\cot A > 1 \Rightarrow (\cot A) (\cot^{3} A) > 1$
$\Rightarrow \tan^{- 1} (\frac{1}{2} \tan 2 A) + \tan^{- 1} (\cot A) + \tan^{- 1} (\cot^{3} A)$
$\begin{array}{l} = \tan^{- 1} (\frac{\tan A}{1 - \tan^{2} A}) + π + \tan^{- 1} (\frac{\cot A + \cot^{3} A}{1 - \cot^{4} A}) \\ = \tan^{- 1} (\frac{\tan A}{1 - \tan^{2} A}) + π + \tan^{- 1} (\frac{\cot A}{1 - \cot^{2} A}) \\ = \tan^{- 1} (\frac{\tan A}{1 - \tan^{2} A}) + π + \tan^{- 1} (\frac{\tan A}{\tan^{2} A - 1}) \\ = π = 4 (π / 4) = 4 \tan^{- 1} (1) \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App