Question

The value of \(\sqrt{\frac{1+sin\ \theta}{1-sin\ \theta}}\) is

• A. secθ+tanθ
• B. cos θ+ sin θ
• C. secθ+cosθ
• D. sin θ+tan θ

Answer 1

The Correct Option is A

To solve the problem, we are given the expression:
\[ \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \]

1. Multiply Numerator and Denominator by Conjugate of Denominator:
Multiply both numerator and denominator by \( \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \cdot \frac{\sqrt{1 + \sin \theta}}{\sqrt{1 + \sin \theta}} \) to rationalize:

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

2. Split the Expression:
\[ \frac{1 + \sin \theta}{\cos \theta} = \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} = \sec \theta + \tan \theta \]

Final Answer:
The value of the expression is \({\sec \theta + \tan \theta} \).