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]
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