Question

If $a = \frac{2 \sin \theta}{1 + \cos \theta + \sin \theta}$, then $\frac{1 + \sin \theta - \cos \theta}{1 + \sin \theta}$ is ________.

• A. $\frac{1}{a}$
• B. $1-a$
• C. $a$
• D. $1+a$

Hint:

Identity used: $1 - \cos^2 \theta = \sin^2 \theta$.

Answer 1

The Correct Option is C

Step 1: Concept
Trigonometric identity simplification and rationalization.
Step 2: Analysis
$a = \frac{2 \sin \theta}{1 + \sin \theta + \cos \theta}$.
Multiply numerator and denominator by $(1 + \sin \theta - \cos \theta)$:
$a = \frac{2 \sin \theta (1 + \sin \theta - \cos \theta)}{(1 + \sin \theta)^2 - \cos^2 \theta}$
Step 3: Calculation
Denominator $= 1 + \sin^2 \theta + 2 \sin \theta - \cos^2 \theta$
$= (1 - \cos^2 \theta) + \sin^2 \theta + 2 \sin \theta$
$= \sin^2 \theta + \sin^2 \theta + 2 \sin \theta = 2 \sin^2 \theta + 2 \sin \theta = 2 \sin \theta (1 + \sin \theta)$.
So, $a = \frac{2 \sin \theta (1 + \sin \theta - \cos \theta)}{2 \sin \theta (1 + \sin \theta)}$.
$a = \frac{1 + \sin \theta - \cos \theta}{1 + \sin \theta}$.
Step 4: Conclusion
The value is equal to $a$.
Final Answer:(C)