Question

How do you use the half angle identity to find exact value of cos^2(pi/12)?

Answer 1

Find `cos^2 (pi/12)`

Ans: `(2 + sqrt3)/4`

Explanation:

Call `cos (pi/12) = cos t`
`cos 2t = cos ((2pi)/12) = cos (pi/6) = sqrt3/2`
a. Apply the double angle identity:
`cos 2t = 2cos^2 t - 1`
`2cos^2 t = 1 + cos 2t = 1 + sqrt3/2 = (2 + sqrt3)/2`
`cos^2 t = cos^2 (pi/12) = (2 + sqrt3)/4`
b. Apply the half angle identity:
`cos^2 (t/2) = (1 + cos t)/2`
In this case cos t = cos (pi/6) = sqrt3/2
`cos^2 (pi/12) = (1 + sqrt3/2)/2 = (2 + sqrt3)/4`