Find the exact value of cos^2(15°)-cos^2(30°)+cos^2(45°)-cos^2(60°)+cos^2(75°)

Answer 1

For this one the 30-60-90° and the 45-45-90° right-angled triangles are your friend.

from those you should know
cos30°= √3/2
sin30° = 1/2

cos60° = 1/2
sin60° = √3/2

sin45° = √2/2
cos45°=√2/2

cos15 = cos(45-30)°
= cos45cos30 + sin45sin30
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4

now do the same for
cos75° = cos(45+30)
= cos45cos30 - sin45sin30
= (√2/2)(√3/2) - (√2/2)(1/2)
= (√6 - √2)/4

We got all our bits and pieces, all we need is the square of those.

I will do one of them:
cos^2 15°
= (√6+√2)^2/16
= (6 + 2√12 + 2)/16
= (8 + 4√3)/16
= (2 + √3)/4

repeat for the other squares, then add them up