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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