Given the following $2$ expressions:

$E_1(x)=\frac{1-cosx}{si{n}^{2}x}$ and $E_2(x)=\frac{1}{1+cosx}$

$($a$)$ Show that $E_1$ is mathematically equivalent to $E_2.$

$($b$)$ Write a code to compute and print out both expressions using the values $x=1\mathrm{.}0,{10}^{-1},{10}^{-2},{10}^{-3},$dots,${10}^{-15}.$ and explain your results.

$($c$)$ What $($if any$)$ precautions should you take to ensure that you have a well$-$defined and robust algorithm?