Question

Prove that \(\dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta} = 2\csc \theta\)

Hint:

Always use identities: \(\sin^2 \theta + \cos^2 \theta = 1\), and simplify numerator and denominator carefully.

Answer 1

LHS: \[ \dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta} \] Take LCM: \[ = \dfrac{\sin^2 \theta + (1 + \cos \theta)^2}{\sin \theta (1 + \cos \theta)} \] Now expand numerator: \[ \sin^2 \theta + 1 + 2\cos \theta + \cos^2 \theta = (\sin^2 \theta + \cos^2 \theta) + 1 + 2\cos \theta = 1 + 1 + 2\cos \theta = 2(1 + \cos \theta) \] So LHS: \[ = \dfrac{2(1 + \cos \theta)}{\sin \theta (1 + \cos \theta)} = \dfrac{2}{\sin \theta} = 2\csc \theta \] LHS = RHS