Question

If \( 2 \sin A = 1 \), then the value of \( \tan A + \cot A \) is :

• A. \( \sqrt{3} \)
• B. \( \frac{4}{\sqrt{3}} \)
• C. \( \frac{\sqrt{3}}{2} \)
• D. \( 1 \)

Hint:

\( \tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta} \). This identity is also useful for faster calculations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Solve for angle A and substitute its value into the required trigonometric expression.
Step 2: Detailed Explanation:
\( 2 \sin A = 1 \implies \sin A = \frac{1}{2} \).
We know \( \sin 30^\circ = \frac{1}{2} \), so \( A = 30^\circ \).
Evaluate: \( \tan 30^\circ + \cot 30^\circ \).
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \quad \text{and} \quad \cot 30^\circ = \sqrt{3} \]
\[ \frac{1}{\sqrt{3}} + \sqrt{3} = \frac{1 + 3}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]
Step 3: Final Answer:
The value is \( \frac{4}{\sqrt{3}} \).