What is (cos 15) ^ 2 – (sin 15)^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