Verified answer

By the half-angle formula for cosine:

cos(x/2) = ±√[(1 + cos x)/2].

(The sign we choose depends on the sign of cos(x/2))

Since cos(5π/12) > 0 because 5π/12 is in Quadrant I, we pick the positive value of the sign. Therefore:

cos(5π/12) = cos[(5π/6)/2]

= √[(1 + cos 5π/6)/2]

= √[(1 - √3/2)/2], since cos(5π/6) = -√3/2

= √[(2 - √3)/4]

= √(2 - √3)/2.

I hope this helps!

First, write 5pi/12 as the sum or difference of two angles for which you can do the trig ratios exactly in your head. 5pi/12 is 75º, so how about 30º + 45º.

Use one of the sum and difference identities:

cos(A + B) = cosAcosB - sinAsinB

cos(5pi/12) = cos(30º + 45º) = cos30ºcos45º - sin30ºsin45º

= √3/2*√2/2 – 1/2*√2/2

= (√6 - √2)/4