Problem $2($USE Matlab$)$

Equation $(1)$ is only valid when $|x|1.$ If you wanted to write your own arctan function, how

would you handle the case $|x|1?$ Please do the following:

Use trig identities to derive this expression relating $arctan(x)$ to $arctan(\frac{1}{x})$ :

$arctan(x)=+--arctan(\frac{1}{x})$

Useful trig identities can be found on the DLMF$,$ in particular equations $4\mathrm{.}23\mathrm{.}9$ and $4\mathrm{.}23\mathrm{.}18.$

Make careful note of the signs of the input and how they affect equation $(2).$ Hand in your

derivation.

Now write a program called $"$my$\_$arctan$"$ which accepts a real input $x.$ Check the value of $x$

and if $|x|1$ then just invoke the series summation directly by calling arctan$\_$series.

Otherwise use equation $(2)$ to transform the input before calling arctan$\_$series. Again, test your

implementation against Matlab's atan$($x$).$