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: Combine $\tan A + \cot A$ into a single identity before finding the angle.
Write both terms with sine and cosine:
\[ \tan A + \cot A = \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} = \frac{\sin^2 A + \cos^2 A}{\sin A \cos A} \]
The numerator $\sin^2 A + \cos^2 A$ is always exactly 1 for any angle, so this simplifies neatly to:
\[ \tan A + \cot A = \frac{1}{\sin A \cos A} \]

Step 2: Find $\sin A$ from the given equation.
\[ 2\sin A = 1 \implies \sin A = \frac{1}{2} \]

Step 3: Find $\cos A$ using the Pythagorean identity, not by naming the angle.
\[ \cos^2 A = 1 - \sin^2 A = 1 - \frac{1}{4} = \frac{3}{4} \]
\[ \cos A = \frac{\sqrt{3}}{2} \]

Step 4: Plug both values into the simplified expression from Step 1.
\[ \sin A \cos A = \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \]
\[ \tan A + \cot A = \frac{1}{\sqrt{3}/4} = \frac{4}{\sqrt{3}} \]

Step 5: Final Answer.
The value of $\tan A + \cot A$ is $\frac{4}{\sqrt{3}}$, so option (B) is correct. \[ \boxed{\dfrac{4}{\sqrt{3}}} \]