cos θ (1-cosθ) = sin θ sinθ
2 2
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.
To solve the given equation cos θ (1 - cosθ) = sin θ sinθ/2, we can use trigonometric identities.
First, let's expand the equation using the identity (1 - cosθ) = sin²θ:
cos θ (1 - cosθ) = sin θ sinθ/2
cos θ sin²θ = sin θ sinθ/2
Next, let's simplify further by dividing both sides of the equation by sinθ:
cos θ sinθ = sinθ/2
Now, let's multiply both sides of the equation by 2 to eliminate the fraction:
2cos θ sinθ = sinθ
Using the double-angle identity sin2θ = 2sinθcosθ, we can substitute it in:
sin2θ = sinθ
Since sinθ is not equal to 0, we can divide both sides by sinθ:
2θ = 1
Finally, divide both sides of the equation by 2 to solve for θ:
θ = 1/2
Therefore, the solution to the equation cos θ (1 - cosθ) = sin θ sinθ/2 is θ = 1/2.