cos θ (1-cosθ) = sin θ sinθ

Question

2 2

Answer 1

We can start by simplifying the left-hand side of the equation using the identity:

cos θ (1 - cos θ) = cos θ - cos² θ

Substituting this back into the original equation, we get:

cos θ - cos² θ = sin² θ

Multiplying both sides by -1, we obtain:

cos² θ - cos θ + sin² θ = 0

Using the Pythagorean identity cos² θ + sin² θ = 1, we can simplify to:

1 - cos θ = 0

Therefore, cos θ = 1, which means that θ = 0 or θ = 2π (or any integer multiple of 2π).

However, we need to check that these values of θ satisfy the original equation.

For θ = 0, the left-hand side of the equation is 0, and the right-hand side is also 0.

For θ = 2π, the left-hand side of the equation is again 0, and the right-hand side is also 0.

Therefore, the solutions to the equation are θ = 0 and θ = 2π.