Using cos15 degrees sq rt of 6 sq rt of 2 4 find the EXAC

Using cos15 degrees = (sq rt of 6 + sq rt of 2) / (4), find the EXACT value for cos7.5 degrees. To get a completely simplified answer, you will have to combine fractions and also rationalize a denominator--twice. Your answer should consist of one fraction with radicals in only the numerator.

Solution

First, let us find cos(15°) using the half-angle formula for cosine:
cos(x/2) = [(1 + cos x)/2].
(Note that the positive value is picked because cos(x/2) > 0)

So:
cos(15°) = [(1 + cos 30°)/2]
= [(1 + 3/2)/2]
= [(2 + 3)/4]
= (2 + 3)/2.

Then, using the half-angle formula for cosine again yields:
cos(7.5°) = [(1 + cos 15°)/2]
= {[1 + (2 + 3)/2]/2}, from above
= {[2 + (2 + 3)]/4}
= [2 + (2 + 3)]/2.