Write the value

Question:

Write the value of $\cos \left(\sin ^{-1} x+\cos ^{-1} x\right),|x| \leq 1$

Solution:

We have

$|x| \leq 1$

$\Rightarrow \pm x \leq 1$

$\Rightarrow x \leq 1$ or $-x \leq 1$

$\Rightarrow x \leq 1$ or $x \geq-1$

$\Rightarrow x \in[-1,1]$

Now,

$\cos \left(\sin ^{-1} x+\cos ^{-1} x\right)=\cos \left(\frac{\pi}{2}\right)$                   $\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right]$

$=0$

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