The value of cos 30 is usually known to all students.

cos 30 = (sqrt 3)/2.

Now we use the formula for cos 2x, which is:

cos 2x = (cos x)^2 - (sin x)^2

=> cos 2x =  2 (cos x)^2 - 1

Now substitute 15 for x here.

We have cos (30) = 2 (cos 15)^2 - 1

=>  (sqrt 3)/2 = 2 (cos 15)^2 - 1

=> 1 + (sqrt 3)/2 = 2 (cos 15)^2

=> (1 + (sqrt 3)/2)/2 = (cos 15)^2

=> cos 15 = sqrt [ (1 + (sqrt 3)/2)/2]

=> cos 15 = sqrt [ 1/2 + (sqrt 3)/4]

You will most probably need a calculator to find the value of the square root here, but you won't need one which has values of cosine stored in it.

Therefore cos 15 can be calculated without a calculator as sqrt [ 1/2 + (sqrt 3)/4]