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:

Use the identity \( \tan A + \cot A = \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} = \frac{\sin^2 A + \cos^2 A}{\sin A \cos A} = \frac{1}{\sin A \cos A} \) to solve it without finding the angle directly if preferred.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Find the specific value of angle \( A \) using the given equation and substitute it into the required expression.
Step 2: Detailed Explanation:
Given: \( 2 \sin A = 1 \implies \sin A = \frac{1}{2} \).
We know from the trigonometric table that \( \sin 30^\circ = \frac{1}{2} \).
So, \( A = 30^\circ \).
Now, we need to evaluate \( \tan A + \cot A \):
\[ \text{Value} = \tan 30^\circ + \cot 30^\circ \]
Substitute the standard values: \( \tan 30^\circ = \frac{1}{\sqrt{3}} \) and \( \cot 30^\circ = \sqrt{3} \).
\[ \text{Value} = \frac{1}{\sqrt{3}} + \sqrt{3} \]
Taking LCM:
\[ \text{Value} = \frac{1 + (\sqrt{3} \times \sqrt{3})}{\sqrt{3}} = \frac{1 + 3}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]
Step 3: Final Answer:
The value is \( \frac{4}{\sqrt{3}} \).