Question

Use trigonometric identities to solve each equation within the given domain.

Answer 1

Recall that

cos²(x) + sin²(x) = 1

Then in the equation

1 - cos(x) = 2 - 2 sin²(x)

we can rewrite as

1 - cos(x) = 2 (1 - sin²(x))

1 - cos(x) = 2 cos²(x)

2 cos²(x) + cos(x) - 1 = 0

Factorize the left side as

(2 cos(x) - 1) (cos(x) + 1) = 0

so that

2 cos(x) - 1 = 0   or   cos(x) + 1 = 0

cos(x) = 1/2   or   cos(x) = -1

On the interval (-π, π) (note that this interval is open, so we don't allow x = π), we have

• cos(x) = 1/2 for x = π/3 and x = -π/3

• cos(x) = -1 for x = π