Differentiate the following with respect to x :

$y=\cos ^{-1}(\sin x)$

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

$\frac{d y}{d x}=\frac{d}{d x}\left(\frac{\pi}{2}-x\right)$

$\frac{d y}{d x}=-1$