Question

How do you express `sin(pi/12) * cos((5 pi)/12 )` without products of trigonometric functions?

Answer 1

`(2 - sqrt3)/4`

Explanation:

On the unit circle, using property of complementary arcs:
`cos ((5pi)/12) = sin (pi/2 + (5pi)/12) = sin ((11pi)/12) = sin pi/12`
Product `P = sin (pi/12).cos ((5pi)/12) = sin^2 (pi/12)`.
Find `sin^2 (pi/12).`
Use trig identity: `cos (pi/6) = 1 - 2sin^2 (pi/12) = sqrt3/2`
`2sin^2 (pi/12) = 1 - sqrt3/2 = (2 - sqrt3)/2`
`P = sin^2 (pi/12) = (2 - sqrt3)/4`