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:

An alternative algebraic method to simplify \(\tan A + \cot A\) is:
\[ \tan A + \cot A = \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} = \frac{\sin^2 A + \cos^2 A}{\sin A \cos A} = \frac{1}{\sin A \cos A} \]
Since \(\sin A = \frac{1}{2}\), then \(\cos A = \sqrt{1 - \sin^2 A} = \frac{\sqrt{3}}{2}\).
Thus:
\[ \frac{1}{\sin A \cos A} = \frac{1}{\frac{1}{2} \times \frac{\sqrt{3}}{2}} = \frac{1}{\frac{\sqrt{3}}{4}} = \frac{4}{\sqrt{3}} \]
This identity-based method is highly reliable and does not require working directly with angles!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a simple trigonometric equation \(2 \sin A = 1\). We need to determine the value of the expression \(\tan A + \cot A\) based on this condition.

Step 2: Key Formula or Approach:
1. Isolate \(\sin A\) from the given equation to find the standard angle \(A\).
2. Once the angle \(A\) is known, substitute it into the expression \(\tan A + \cot A\) and evaluate using standard trigonometric values.
3. Alternatively, we can use trigonometric identities to express \(\tan A + \cot A\) in terms of \(\sin A\) and \(\cos A\).

Step 3: Detailed Explanation:
1. Solve the given equation for \(\sin A\):
\[ 2 \sin A = 1 \implies \sin A = \frac{1}{2} \]
2. Identify the standard angle \(A\) in the first quadrant for which \(\sin A = \frac{1}{2}\):
\[ A = 30^\circ \]
3. Substitute \(A = 30^\circ\) into the target expression \(\tan A + \cot A\):
\[ \tan A + \cot A = \tan 30^\circ + \cot 30^\circ \]
4. Substitute the standard values of \(\tan 30^\circ\) and \(\cot 30^\circ\):
We know that \(\tan 30^\circ = \frac{1}{\sqrt{3}}\) and \(\cot 30^\circ = \sqrt{3}\).
\[ \tan A + \cot A = \frac{1}{\sqrt{3}} + \sqrt{3} \]
5. Simplify the expression by taking a common denominator:
\[ \tan A + \cot A = \frac{1 + (\sqrt{3} \times \sqrt{3})}{\sqrt{3}} = \frac{1 + 3}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]
The value of the expression is \(\frac{4}{\sqrt{3}}\).

Step 4: Final Answer:
The value of \(\tan A + \cot A\) is \(\frac{4}{\sqrt{3}}\), which corresponds to option (B).