Here we have to use the expression for the cosine of 2x in terms of the cosine of x. We know that cos 2x is given by the expression (cos x) ^2 – (sin x) ^2.

Now we are given (cos 15) ^ 2 – (sin 15) ^2 as the expression to be evaluated in the question.

=> (cos 15) ^ 2 – (sin 15) ^2 = cos (15*2)

=> cos 30

we also know that cos 30 = sqrt 3/ 2

**Therefore (cos 15) ^ 2 – (sin 15) ^2 = sqrt 3 /2**

We could also solve this problem as a difference of squares:

a^2 - b^2 = (a-b)(a+b)

a = cos 15 and b = sin 15

(a-b)(a+b) = (cos15 - sin15)(cos15 + sin 15)

We could also consider 15 degrees as the half of 30 degrees.

cos 15 = cos 30/2

(cos 15)^2 = (cos 30/2)^2

(cos 30/2)^2 = (1+cos30)/2

(cos 30/2)^2 = (2+sqrt3)/4

(sin 30/2)^2 = (2-sqrt3)/4

(cos 15)^2 - (sin 15)^2 = (cos 30/2)^2 - (sin 30/2)^2

(cos 30/2)^2 - (sin 30/2)^2 = (2+sqrt3-2+sqrt3)/4

We'll eliminate and combine like terms:

(cos 15)^2 - (sin 15)^2 = 2sqrt3/4

**(cos 15)^2 - (sin 15)^2 = sqrt3/2**

To find cos^2 15 -sin^2 15 .

We know that cos^x+sin ^2 x = 1.

Therefore cos^x = 1-sin^2x.

Therefore cos^2 15 - sin^215 = cos^215 - (1-sin^215 ) = 2cos2^15 -1 = cos2*15 = cos30 = (sqrt3)/2