Question

\(\frac{sinθ}{1+cosθ}=\)

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

Answer 1

The Correct Option is C

Step 1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of \( 1 + \cos\theta \) is \( 1 - \cos\theta \). Multiply both numerator and denominator by \( 1 - \cos\theta \):

\[ \frac{\sin\theta}{1 + \cos\theta} = \frac{\sin\theta (1 - \cos\theta)}{(1 + \cos\theta)(1 - \cos\theta)}. \]

Step 2: Simplify the denominator using the difference of squares formula.

The denominator becomes:

\[ (1 + \cos\theta)(1 - \cos\theta) = 1 - \cos^2\theta. \]

Using the Pythagorean identity \( 1 - \cos^2\theta = \sin^2\theta \), the expression becomes:

\[ \frac{\sin\theta (1 - \cos\theta)}{\sin^2\theta}. \]

Step 3: Simplify the fraction.

Cancel one \( \sin\theta \) from the numerator and denominator:

\[ \frac{\sin\theta (1 - \cos\theta)}{\sin^2\theta} = \frac{1 - \cos\theta}{\sin\theta}. \]

Final Answer: The simplified value of \( \frac{\sin\theta}{1 + \cos\theta} \) is \( \mathbf{\frac{1 - \cos\theta}{\sin\theta}} \), which corresponds to option \( \mathbf{(3)} \).