Question:
Differentiate the following with respect to $x$ :
$\cos -1(\sin x)$
Solution:
$y=\cos ^{-1}(\sin x)$
Function is defined for all $x$
$y=\cos ^{-1}\left(\cos \left(\frac{\pi}{2}-x\right)\right)$
$y=\frac{\pi}{2}-x$
Differentiating w.r.t $x$ we get
$\frac{d y}{d x}=\frac{d}{d x}\left(\frac{\pi}{2}-x\right)$
$\frac{d y}{d x}=-1$
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