Could someone help me out with this?

Let $a$ , $b$ and $c$ be positive real numbers such that $\theta=\tan^{-1} \sqrt{\frac {a(a+b+c)}{bc}} + \tan^{-1} \sqrt{\frac {b(a+b+c)}{ca}} + \tan^{-1} \sqrt{\frac {c(a+b+c)}{ab}}$ Then find the value of $\tan \theta$ .

I'm thinking of using that identity $\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\frac{xy}{1-xy}\right)$ , but I can't get anywhere with it. Is there any way of solving this without that identity?

#Geometry #Trigonometry #InverseTrigonometricFunctions #JEE

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

A possible way is :

Let $\displaystyle \alpha =\arctan \left( \sqrt{\frac{a(a+b+c)}{abc}}\right) = \arctan \left(a \sqrt{\frac{a+b+c}{abc}}\right) =\arctan ( ak)$ . Similarly let $\displaystyle \beta = \arctan (bk)$ and $\gamma = \arctan(ck)$ , where $\displaystyle k = \sqrt{\frac{(a+b+c)}{abc}}$ .

So, $\displaystyle \theta = \alpha + \beta + \gamma \Rightarrow \tan \theta = \tan (\alpha + \beta + \gamma)$ .

$\displaystyle \begin{array}{c}\\ & = \frac{\tan \alpha + \tan \beta + \tan \gamma - \tan \alpha \tan \beta \tan \gamma}{\tan \alpha \tan \beta + \tan \beta \tan \gamma + \tan \gamma \tan \alpha } \\ & = \frac{(a+b+c)k -abck^3}{(ab+bc+ca)k^2} \\ & = \frac{k\left( (a+b+c) - abc \left(\frac{a+b+c}{abc} \right) \right) }{(ab+bc+ca)k^2} \\ & = 0 \\ \end{array}$ .

$\displaystyle \therefore\boxed{ \tan \theta = 0}$ .

Sudeep Salgia - 6 years, 4 months ago

Ohh okay. Is there no way other than using the $\tan(\alpha+\beta+\gamma)$ identity?

Omkar Kulkarni - 6 years, 4 months ago

I cannot of think about such a method now. I'll post if I get one.

Sudeep Salgia - 6 years, 4 months ago

@Sudeep Salgia – Okay thanks!

Omkar Kulkarni - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$