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  4. If cos-1 x + cos-1 y + cos-1z = 3π , then xy + yz + zx is

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Q. If $\cos^{-1} x +\cos^{-1} y +\cos^{-1}z = 3\pi $, then $xy + yz + zx$ is

COMEDKCOMEDK 2011Inverse Trigonometric Functions

Solution:

$ \cos^{-1} x +\cos^{-1} y +\cos^{-1}z = 3\pi$
We know that $ 0\le\cos ^{-1}x \le\pi$
$ \Rightarrow \cos ^{-1} x =\cos ^{-1}y =\cos ^{-1}z =\pi$
$\Rightarrow x = y=z= -1$
Now, $ xy+yz+zx$
$ = \left(-1\right)\left(-1\right)+\left(-1\right)\left(-1\right)+\left(-1\right)\left(-1\right)=3$