Question

Prove that sin^2θ/(1-cosθ) = 1 + cosθ.

Answer 1

To prove the identity sin^2θ/(1-cosθ) = 1 + cosθ, we use the Pythagorean identity to express sin^2θ as (1 - cos^2θ), factor, and then cancel common terms, ultimately showing both sides are equal.

Step-by-step explanation:

To prove that sin^2θ/(1-cosθ) = 1 + cosθ, let's manipulate the left side of the equation using trigonometric identities.

1. Express sin^2θ as (1 - cos^2θ) using the Pythagorean identity sin^2θ + cos^2θ = 1.
2. Place (1 - cos^2θ) in place of sin^2θ in the numerator to get ((1 - cos^2θ)/(1 - cosθ)).
3. Factor the numerator as (1 - cosθ)(1 + cosθ).
4. Cancel out the (1 - cosθ) term in the numerator and denominator.
5. What remains is 1 + cosθ, which proves the initial equation.