If the equation is
- sin²(x) = cos(2x)
rewrite the left side using the Pythagorean identity,
sin²(x) = 1 - cos²(x)
and the right side using the double angle identity for cosine,
cos(2x) = 2 cos²(x) - 1
Then the equation transforms to
- (1 - cos²(x)) = 2 cos²(x) - 1
Solve for cos²(x) :
-1 + cos²(x) = 2 cos²(x) - 1
cos²(x) = 0
Take the square root of both sides to get
cos(x) = 0
Solve for x :
x = cos⁻¹(0) + 2nπ or x = cos⁻¹(0) - π + 2nπ
x = π/2 + 2nπ or x = -π/2 + 2nπ
where n is any integer. We get solutions in the interval [-π, π] when n = 0, giving the solutions
x = π/2 or x = -π/2
