For this question, we define the tangent function as $tan(x)=\frac{sin(x)}{cos(x)}$ on $\frac{R}{\frac{?}{?}}\{(2k+1)\frac{}{2}$:kinZ$\}.$

$($a$)$ Prove that $tan(x)$ is differentiable on its domain and determine its derivative. $($Again$,$ simplify as much as you can.$)$

$($b$)$ Define $arctan$:$R(-\frac{}{2},\frac{}{2})$ by $arctan(y)=x$ if and only if $tan(x)=y.$ Prove that arctan is differentiable on $R$ and determine its derivative. Simplify your final answer as much as you can.