(1) Recall the angle sum identity for cosine:
cos(x + y) = cos(x) cos(y) - sin(x) sin(y)
So
cos(π/3) cos(5π/6) - sin(π/3) sin(5π/6) = cos(π/3 + 5π/6)
= cos(7π/6)
= -√(3)/2
(2) Recall the half-angle identity:
cos²(x) = (1 + cos(2x))/2
Since π/2 < 7π/12 < π, you should expect cos(7π/12) < 0. So solving for cos(x) above, we have
cos(x) = -√[(1 + cos(2x))/2]
Then
cos(7π/12) = -√[(1 + cos(7π/6))/2]
= -√[(1 - √(3)/2)/2]
= -√[1/2 - √(3)/4]
You could stop there, or continue and try to de-nest the radicals to end up with
= (√(2) - √(6))/4
(3) Recall the angle sum formula for tangent:
tan(x - y) = (tan(x) - tan(y))/(1 + tan(x) tan(y))
Then
(tan(122°) - tan(32°))/(1 + tan(122°) tan(32°)) = tan(122° - 32°)
= tan(90°)
which is undefined, since tan(x) = sin(x)/cos(x), and cos(90°) = 0.
(4) Recall the double angle identity for sine:
sin(2x) = 2 sin(x) cos(x)
Then
2 sin(15°) cos(15°) = sin(30°)
= 1/2
(5) Use the double angle identity for cosine:
cos(2x) = 2 cos²(x) - 1
Then
12 cos²(θ) - 6 = 6 (2 cos²(θ) - 1)
= 6 cos(2θ)
(6) Use the sine double angle identity mentioned in (4):
7 cos(4x) sin(4x) = 7/2 (2 cos(4x) sin(4x))
= 7/2 sin(8x)
Consult the other question you commented on [19739123 in the URL] for details on (7) and (8).
