Question

\(\frac{sinθ}{1+cos θ}=\)

• A. \(\frac{1+cosθ}{sin θ}\)
• B. \(\frac{1-cosθ}{cos θ}\)
• C. \(\frac{1-cosθ}{sin θ}\)
• D. \(\frac{1+sinθ}{cos θ}\)

Answer 1

The Correct Option is C

We are given the expression: \[ \frac{\sin \theta}{1 + \cos \theta} \] To simplify this expression, multiply both the numerator and the denominator by \( 1 - \cos \theta \): \[ \frac{\sin \theta}{1 + \cos \theta} \cdot \frac{1 - \cos \theta}{1 - \cos \theta} = \frac{\sin \theta (1 - \cos \theta)}{(1 + \cos \theta)(1 - \cos \theta)} \] Using the identity \( (a + b)(a - b) = a^2 - b^2 \), we simplify the denominator: \[ (1 + \cos \theta)(1 - \cos \theta) = 1^2 - \cos^2 \theta = \sin^2 \theta \] Thus, the expression becomes: \[ \frac{\sin \theta (1 - \cos \theta)}{\sin^2 \theta} = \frac{1 - \cos \theta}{\sin \theta} \] Therefore, the simplified expression is: \[ \frac{1 - \cos \theta}{\sin \theta} \]

The correct option is (C): \(\frac{1-cosθ}{cos θ}\)