Find the exact value by using a halfangle identity cos 75 de

Find the exact value by using a half-angle identity. cos 75 degree A. -1/2 (Squareroot 2 - Squareroot 3) B. 1/2(Squareroot 2 - Squareroot 3) C. 1/2 (Squareroot 2 + Squareroot 3) D. -1/2 (Squareroot 2 + Squareroot 3)

Solution

Note that, by the half-angle formula for cosine:
cos(x/2) = ±[(1 + cos x)/2].
(The sign selected depends on the sign of cos(x/2).)

Since 75° is in Quadrant I, we see that cos(75°) > 0. Thus, pick the positive sign.
cos(x/2) = [(1 + cos x)/2].

By letting x/2 = 75° ==> x = 150°:
cos(75°) = [(1 + cos 150°)/2]
= [(1 - 3/2)/2], since cos(150°) = -3/2
= [(2 - 3)/4]
= (2 - 3)/2. (option B)

I hope this helps!