How do you find exact value of ##cos (pi/ 4)##?

Let's first convert to degrees.

The conversion ratio between degrees and radians is ##180/pi##.

##pi/4 xx 180/pi = 45^@##

Now, we must think of the special triangle that contains a ##45^@## angle. This would be the following triangle.

The definition of cos is adjacent/hypotenuse. Therefore, our ratio is ##1/sqrt(2)##. However, often, teachers want the denominator to be rationalized, or be a rational number. Since the expression is a monomial, we multiply the entire expression by the denominator.

##1/sqrt(2) xx sqrt(2)/sqrt(2)##

=##sqrt(2)/2##

Hopefully this helps.