How can the value of cos 15 be found without the use of a calculator. Which formula has to be used?

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]