Find all solutions of the equation in the interval [0,2pi) cos2x+2cosx=-1

Answer 1

cos 2x + 2 cosx = -1

There are various ways that cos 2x can be expressed. As the other term is a cosine term, you choose the expression involving cosine, whish is 2 cos²x - 1

Thus,

2 cos²x - 1 + 2 cosx = -1

2 cos²x + 2 cosx = 0

2 cos x (cos x + 1) = 0

cos x = 0, -1

We are looking for solutions in [0, 2π),

cos x = 0 at π/2 and 3π/2

cos x = -1 at π

The solution is

(π/2, π, 3π/2)

Answer 2

The substitution cos 2x = 2 cos²x -1 will convert this equation to a quadratic is cos x which can be solved by the quadratic formula.