Question

All of the questions in the picture

Answer 1

(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).