$($a$)$ Define the function $arcsin$:$[-1,1][-\frac{}{2},\frac{}{2}]$ by $arcsin(y)=x$ if and only if $sin(x)=y.$ Prove that arcsin is differentiable on $-1,1$ and determine its derivative. Simplify your final answer as much as possible.

$($b$)$ Define the function $arccos$:$[-1,1][0,]$ by $arccos(y)=x$ if and only if $cos(x)=y.$ Prove that arccos is differentiable on $-1,1$ and determine its derivative. Simplify your final answer as much as possible.