My math teacher gave us some trig questions to do but I dont

My math teacher gave us some trig questions to do but I don\'t know how to do these ones. Could you show all of your steps in your work please

1. Using a double angle formula for cosine, derive the half angle formula for sine and cosine.

sin x/2 = +/- 1-cosx/2 cos x/2 = +/- 1+cosx/2

2. Prove sinx + siny = 2sin(x+y/2) cos(x-y/2)

3. Angle x lies in the third quadrant, and tanx=7/24. Determine an exact value for cos2x and sin2x

4. Find an exact value for cos7/24. Solve it using 7/24 = /6 + /8 (You will have to use a half angle formula for this).

5. Using the quotent identity (tanx=sinx/cosx), show that tan(x+y) = tanx + tany/1-tanxtany

Solution

1) As sin^2 x + cos^2 x = 1

So sin (x/2) = sqrt[1 - cos^2 (x/2)]

And cos2x = 2*cos^2 x - 1

So cos(x/2) = sqrt[(cos x + 1)/2]

2) As sinX + sinY = 2sin[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]

sinx + siny = 2sin(x+y/2) cos(x-y/2)

3) tanx=7/24

cos2x = (1-tan²(x))/(1+tan²(x)) = (1 - 49/576)/(1 + 49/576) = 527/625 = -0.8432

sin2x = sqrt1[ - cos^2 (2x)] = -0.5376

4) cos(A + B) = cos A cos B sin A sin B

cos7/24 = cos(/6 + /8) = cos/6 cos/8 sin/6 sin/8 = 0.866*0.924 - 0.5*0.383 = 0.6087

5) tan(x+y) = tanx + tany/1-tanxtany

LHS = tan(x+y) = (tan x + tan y)/(1 tan x tan y) = RHS