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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

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a

π4

b

π2

c

π

d

none of these

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detailed solution

Correct option is C

The given expression can be written astan−1⁡aa+b+cabc+tan−1⁡ba+b+cabc+tan−1⁡ca+b+cabc=tan−1⁡(ay)+tan−1⁡(by)+tan−1⁡(cy)where, y=a+b+cabc=tan−1⁡ay+by+cy−abcy31−aby2−bcy2−acy2=tan−1⁡ya+b+c−abcy21−y2(ab+bc+ca)=tan−1⁡0=0 or π

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