The domain set of definition of the function $f(x)=\sqrt{\cos(\sin x)}+\sin^{-1}\left(\frac{1+x^2}{2x}\right)$ is

$x=±1$

cos (sin x) ≥ 0 for all x ; x ≠ 0

Hence cos (sin x) > 0 for all x

$-\frac{π}{2}≤\sin^{-1}\left(\frac{1+x^2}{2x}\right)≤+\frac{π}{2}$

i.e. $-1≤\frac{1+x^2}{2x}⇒-2x≥(1+x^2)⇒(1+x)^2≤0⇒x=-1$

The other side $(x>0),1+x^2≤2x⇒(x-1)^2≤0⇒x=1$

The function is defined only for the two values x = 1 and x = −1.