First slide

Properties of ITF

Question

The value of $\tan^{- 1} \sqrt{\frac{a (a + b + c)}{bc}} + \tan^{- 1} \sqrt{\frac{b (a + b + c)}{ca}} + \tan^{- 1} \sqrt{\frac{c (a + b + c)}{ab}}$ is

Moderate

A

$\frac{π}{4}$
B

$\frac{π}{2}$
C

$π$
D

none of these

Solution

The given expression can be written as
$\tan^{- 1} (a \sqrt{\frac{a + b + c}{abc}}) + \tan^{- 1} (b \sqrt{\frac{a + b + c}{abc}}) + \tan^{- 1} (c \sqrt{\frac{a + b + c}{abc}})$
$= \tan^{- 1} (ay) + \tan^{- 1} (by) + \tan^{- 1} (cy)$
where, $y = \sqrt{\frac{a + b + c}{abc}}$
$\begin{matrix} = \tan^{- 1} (\frac{ay + by + cy - {abcy}^{3}}{1 - {aby}^{2} - {bcy}^{2} - {acy}^{2}}) \\ = \tan^{- 1} [y (\frac{a + b + c - {abcy}^{2}}{1 - y^{2} (ab + bc + ca)})] \\ = \tan^{- 1} 0 = 0 or π \end{matrix}$

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