Use half angle identities to solve trig equations

Question

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

1. cos^2 x + 4 sin^2 x/2 = 1
2. cos x/2 - sin x = 0 ( hint: sin^2 x + cos^2 x = 1)
3. cos^2 x = sin^2 x/2

Answer 1

I will not solve these for you, but I will tell you how to do them.

In each case use of the half-angle formula and sin² + cos² = 1 will get you a quadratic equation in one of the trig functions, e.g in the first one:

cos²x + 2(1-cos x) =2 becomes cos²x - 2 cos x +1=0.

With this hint and some persevereance you should be able to solve all of these!