How to find the exact value of cos ( 5 π ) 6 ?

Well, the odd multiples of the cosine ${\displaystyle pI}$ is always -1 hence ${\displaystyle \cos \left(5\pi \right)=-1}$ and
${\displaystyle \frac{\cos \left(5\pi \right)}{6}=-\frac{1}{6}}$
Although if you mean ${\displaystyle \cos \left(\frac{5\pi }{6}\right)}$,
it can be found by using identity ${\displaystyle \cos (\pi -x)=-\cos x}$
and ${\displaystyle \cos \left(\frac{5\pi }{6}\right)}$
= ${\displaystyle \cos (\pi -\frac{\pi }{6})}$
= ${\displaystyle -\cos \left(\frac{\pi }{6}\right)}$
= ${\displaystyle -\frac{\sqrt{3}}{2}}$

There are two ways to complete the task, but if you remember the unit circle, you can complete them more quickly.
Convert ${\displaystyle \frac{5\pi }{6}}$ to angle degrees by using the equation:
${\displaystyle rad\cdot \left(\frac{180}{\pi }\right)=degrees}$
${\displaystyle \frac{5\pi }{6}\cdot \frac{180}{\pi }={150}^{\circ }}$
And we can figure out that the reference angle for ${\displaystyle {150}^{\circ }}$ is ${\displaystyle {30}^{\circ }}$.
${\displaystyle {\cos 30}^{\circ }=\frac{\sqrt{3}}{2}}$
Since ${\displaystyle {150}^{\circ }}$ is in the 2nd quadrant, we know cosine is negative.
${\displaystyle {\cos 30}^{\circ }=\cos \left(\frac{5\pi }{6}\right)=-\frac{\sqrt{3}}{2}}$