Use an appropriate halfangle formula to find the exact value

Use an appropriate half-angle formula to find the exact value of cos(5 pi/8). Show your work.

Solution

cos(5pi/8)

cos2x = 2cos^2x -1

cosx = sqrt[ (1 +cos2x)/2 ]

cos(5pi/8) = - sqrt[(1 +cos(5pi/4) )/2 ]

cos5pi/4 = cos(pi +pi/4) = -cospi/4 = -1/sqrt2

cos(5pi/8) = - sqrt[(1 -1/sqrt2 )/2 ]

= -sqrt[(sqrt2 - 1)/2sqrt2]