Recall the half-angle identity:
cos²(x) = 1/2 (1 + cos(2x))
Let x = 75°, so that 2x = 150°. Then
cos²(75°) = 1/2 (1 + cos(150°))
You might already be aware that cos(150°) = -√3/2, so
cos²(75°) = 1/2 (1 - √3/2)
cos²(75°) = 1/2 - √3/4
cos²(75°) = (2 - √3)/4
But this is the square of the number we want, which we solve for by taking the square root of both sides. This introduces a second solution, however:
cos(75°) = ± √[(2 - √3)/4]
cos(75°) = ± √(2 - √3)/2
75° falls between 0° and 90°, and you should know that cos(x) is positive for x between these angles. This means cos(75°) must be positive, so we pick the positive root:
cos(75°) = √(2 - √3)/2
