There are two ways to evaluate cos 4? that will both give the answer of 1. The best ways to evaluate involve the periodicity of the cosine function and the trigonometric addition formula for cosine.
First, the cosine function has a period of 2?, meaning that cos x = cos (x + 2?). Using this formula, we know that cos 0 = cos 2? = cos 4?. Recalling that cos 0 = 1, we know that cos 4? = cos 0 = 1.
In cases of cos 2n? or sin 2n?, where n is a whole number, this is a simple process. However, this process does not always involve simple terms like this. This is when the trigonometric addition formula for cosine is helpful.
First, suppose that cos z needs to be evaluated, and z = x + y, where x, y and z are real numbers.
Then, cos(x + y) = (cos x)(cos y) -(sin x)(sin y)
Using 2? as both x and y to make z = 4?,
cos 4? = cos (2? + 2?) = (cos 2?)(cos 2?) - (sin 2?)(sin 2?)
Now, recalling that cos 2? = 1 and sin 2? = 0,
cos 4? = (1) (1) - (0) (0) = 1.
While this method involves much more calculation, it works for any angle.