Answer
a) The value is $0$.
b) The value is $0$.
Work Step by Step
(a)
It is given that $\cot \left( t \right)=\frac{\cos \left( t \right)}{\sin \left( t \right)}$
So,
$\cot \left( \frac{\pi }{2} \right)=\frac{\cos \left( \frac{\pi }{2} \right)}{\sin \left( \frac{\pi }{2} \right)}$
In the unit circle, the point corresponding to $t=\frac{\pi }{2}$ has the coordinates $\left( 0,1 \right)$. Therefore, use $x=0$ and $y=1$ such that,
$\sin \left( \frac{\pi }{2} \right)=y=1$ and $\cos \left( \frac{\pi }{2} \right)=x=0$
So,
$\begin{align}
& \cot \left( \frac{\pi }{2} \right)=\frac{\cos \left( \frac{\pi }{2} \right)}{\sin \left( \frac{\pi }{2} \right)} \\
& =\frac{0}{1} \\
& =0
\end{align}$
Thus, the value of the trigonometric function is $0$.
(b)
We know that the periodic properties of sine and cosine functions are,
$\tan \left( t+\pi \right)=\tan \left( t \right)$ and $\text{cot}\left( t+\pi \right)=\cot \left( t \right)$.
So,
$\begin{align}
& \cot \left( \frac{15\pi }{2} \right)=\cot \left( \frac{\pi }{2}+7\pi \right) \\
& =\cot \left( \frac{\pi }{2} \right)
\end{align}$
Now put $\cot \left( \frac{\pi }{2} \right)=0$. So,
$\cot \left( \frac{15\pi }{2} \right)=0$
Thus, the value of the trigonometric function $\cot \left( \frac{15\pi }{2} \right)$ is $0$.
