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  4. If sec-1 √1+x2 + cosec -1 (√1+y2/y) + cot-1 (1/z) =π then x + y + z is equal to

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Q. If $\sec^{-1} \sqrt{1+x^{2}} + cosec ^{-1} \frac{\sqrt{1+y^{2}}}{y} + \cot^{-1} \frac{1}{z} =\pi $ then x + y + z is equal to

Inverse Trigonometric Functions

Solution:

Given : $\sec^{-1} \sqrt{1+x^{2}} + cosec ^{-1} \frac{\sqrt{1+y^{2}}}{y} + \cot^{-1} \frac{1}{z} =\pi $
$ \Rightarrow \tan^{-1} x +\tan^{-1} y + \tan^{-1}z =\pi $
$\Rightarrow \tan^{-1} \left(\frac{x+y+z-xyz}{1-xy-yz-zx}\right) =\pi$
$ \Rightarrow \frac{x+y+z-xyz }{1-xy-yz-zx} = \tan\pi= 0$
$ \Rightarrow x + y + z = xyz$