Use half angle identities to solve trig equations

Question

Use half angle identities to solve trig equations 

Solve the given equations for the values of x in the interval [0, 2pi]

1. cos x/2 - sin x = 0 ( hint: sin^2 x + cos^2 x = 1)
2. cos^2 x = sin^2 x/2

Answer 1

Cos(x/2) - Sin(x) = 0

√[(1+Cosx)/2] = Sinx

1+Cosx = 2Sin²

1+Cosx = 2(1-Cos²x)

1+Cosx = 2-2Cos²x

2Cos²x + Cosx -1 = 0

2Cosx - 1)(Cosx +1) = 0

Set each factor= 0

Cosx = 1/2 , Cosx= -1

x= π/3 or 60° and π