If $$r=x+y+z$$ and $$tan$$−$$1xryz+$$$$tan$$−$$1yrzx+$$$$tan$$−$$1zrxy=k$$$$π$$then the value of k is

Let $$x=y=z=1$$. ⇒$$r=3$$ Then the equation becomes $$tan$$$$-13+$$$$tan$$$$-13+$$$$tan$$$$-13=k$$π ⇒π$$3+$$π$$3+$$π$$3=k$$π ⇒π$$=k$$π ⇒$$k=1$$

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Question

If $r = x + y + z$ and $\tan^{- 1} \sqrt{\frac{x r}{y z}} + \tan^{- 1} \sqrt{\frac{y r}{z x}} + \tan^{- 1} \sqrt{\frac{z r}{x y}} = k π$ then the value of k is

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Solution

$L e t x = y = z = 1 . \Rightarrow r = 3 T h e n t h e e q u a t i o n b e c o m e s \tan^{- 1} \sqrt{3} + \tan^{- 1} \sqrt{3} + \tan^{- 1} \sqrt{3} = k π \Rightarrow \frac{π}{3} + \frac{π}{3} + \frac{π}{3} = kπ \Rightarrow π = kπ \Rightarrow k = 1$

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