Question

Prove that sin θ / (1 + cos θ) + (1 + cos θ) / sin θ = 2 cosec θ.

Solution

1. Take a common denominator on the left side: [sin^2 θ + (1 + cos θ)^2] / [sin θ (1 + cos θ)].
2. Expand the numerator: sin^2 θ + 1 + 2 cos θ + cos^2 θ = (sin^2 θ + cos^2 θ) + 1 + 2 cos θ = 2 + 2 cos θ = 2(1 + cos θ).
3. So the left side = 2(1 + cos θ) / [sin θ (1 + cos θ)] = 2 / sin θ.
4. Since 1 / sin θ = cosec θ, the left side = 2 cosec θ, which is the right side.