Find the exact value of the trigonometric function cos (9pi)/4.

Rewriting the given trigonometric function,
${\displaystyle \frac{\cos \left(9\pi \right)}{4}=\cos \left(\frac{8\pi +\pi }{4}\right)}$
${\displaystyle =\cos (\frac{8\pi }{4}+\frac{\pi }{4})}$
${\displaystyle =\cos (2\pi +\frac{\pi }{4})}$
We know that ${\displaystyle (2\pi +\theta )}$ always lies in first quadrant and in first quadrant all trigonometric functions are positive.
So applying above rule for the given trigonometric function, we get
${\displaystyle \cos (2\pi +\frac{\pi }{4})=\cos \left(\frac{\pi }{4}\right)}$
${\displaystyle =\frac{1}{\sqrt{2}}}$
Hence, exact value of the given trigonometric function is ${\displaystyle \frac{1}{\sqrt{2}}}$