Q$3.$ Create a NEW Jupyter Notebook file with the name $"$PT$1$Q$3".$ Copy the content of the source file in Q$2$ to a source file in this project. Modify the program developed for Q$2$ to perform the following tasks:

$-$ Rewrite the function showMenu$()$ such that if the user enters $\text{'}\backslash ($ c $\backslash )\text{'},$ it will display "Please enter value of $\backslash ($ x $\backslash )$ : $".$ The function calcos$()$ is called with a float data type number and an integer value from functions $\backslash (\backslash$boldsymbol$\{$g e t$\}\_\{\backslash$mathbf$\{$t$\}\}\backslash$mathbf$\{$x$\}()\backslash )$ and $\backslash (\backslash$boldsymbol$\{$g e t$\}\backslash$mathbf$\{$k$\}()\backslash )$ respectively as the input parameter values. The function calcos$()$ returns a value of type float, which is the approximated cosine function value calculated by using equation $(1)$ on page $1.$ This returned result should be printed by showMenu $().$ Inside the function calCos$(),$ you must call the function calTerm$()$ in Q$1$ to compute the approximated cosine function value.

$-$ You may use math. $\backslash (\backslash$cos $(\backslash ))$ library function to compare with your result.

$-$ By default, print$()$ with $\backslash (\backslash \{$: g$\backslash \}\backslash )$ format will only show a floating$-$point number with $6$ significant digits. Hence, the result under different $\backslash ($ k $\backslash )$ may be the same owing to inadequate precision. To see the difference, you need to show a floating$-$point number with more significant figures. Modify your program such that you can show the result as a $15-$digit number. This is achieved by using the following statement to show the value of variable $\backslash (\backslash$mathbf$\{$f$\}\backslash )$ :

$\backslash [$

$\backslash$text $\{$ print$("\backslash \{$:$\mathrm{.}15$f$\backslash \}".$ format$($f$))\}$

$\backslash ]$

$-$ Finally, implement a new better function calCosNew $()$ which performs the same task of calCos$()$ without using the function calTerm$().$ You may use Horner's rule $($only $\backslash ($ n$+1\backslash )$ multiplications are needed $)$ for polynomial evaluation, i$.$e$.$

$\backslash [$

P$($x$)=$a$\_\{$n$\}$ x$^\{$n$\}+$a$\_\{$n$-1\}$ x$^\{$n$-1\}+\backslash$cdots$+$a$\_\{0\}=\backslash$left$(\backslash$left$(\backslash$cdots$\backslash$left$($a$\_\{$n$\}$ x$+$a$\_\{$n$-1\}\backslash$right$)$ x$+$a$\_\{$n$-2\}\backslash$right$)$ x$+\backslash$cdots$+$a$\_\{0\}\backslash$right$.$

$\backslash ]$

$-$ The sample outputs given below should be realised by your program on executing it$.$ Notice that the character$($s$)$ following a colon $($: $)$ is$/$are entered by the user.

$```$

$[$t$]$ Get a term in cosine function

$[$a$]$ Get the terms in cosine function

$[$c$]$ Get cosine of an angle in radian

$[$q$]$ Quit

State your choice $(\}$t$,$a$,$c$\backslash$mathrm$\{,$ or q to quit$)$: c

Please enter value of x: $6$

$-3\mathrm{.}1416$