Question

How do you simplify `cos²(pi/12)-sin²(pi/12)`?

Answer 1

Find `cos^2 (pi/12) - sin^2 (pi/12)`
Ans:`(sqrt3)/2`

Explanation:

Call `cos (pi/12) = cos t` --> `cos 2t = cos pi/6 = sqrt3/2`
Call `sin (pi/12) = sin t`
Use the trig identity: `cos 2t = 2cos^2t - 1` and `cos 2t = 1 - 2sin^2 t`
`cos 2t = sqrt3/2 = 2cos^2 t - 1 -> cos^2 t = (2 + sqrt3)/4`

`cos 2t = sqrt3/2 = 1 - 2sin^2 t -> sin^2 t = (2 - sqrt3)/4`

`cos^2 t - sin^2 t = (2 + sqrt3 - 2 + sqrt3)/4 = (2sqrt3)/4 = sqrt3/2`