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

To solve for the value of \( \tan A + \cot A \) given that \( 2 \sin A = 1 \), follow these step-by-step calculations:

1. First, determine the value of \( \sin A \) from the equation \( 2 \sin A = 1 \): 
   \(\sin A = \frac{1}{2}\)
2. The angle \( A \) for which \( \sin A = \frac{1}{2} \) in the first quadrant is \( 30^\circ \) or \( \frac{\pi}{6} \) radians.
3. Calculate \( \tan A \): 
   \(\tan A = \frac{\sin A}{\cos A}\) 
   Given \( \sin A = \frac{1}{2} \), we use the identity \( \cos^2 A = 1 - \sin^2 A \) to find: 
   \(\cos^2 A = 1 - \left(\frac{1}{2}\right)^2 = \frac{3}{4}\) 
   Thus, \(\cos A = \frac{\sqrt{3}}{2}\) (considering only positive values in the first quadrant). 
   Now substitute to find \( \tan A \): 
   \(\tan A = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\)
4. Calculate \( \cot A \): 
   \(\cot A = \frac{1}{\tan A} = \sqrt{3}\)
5. Find the sum \( \tan A + \cot A \): 
   \(\tan A + \cot A = \frac{1}{\sqrt{3}} + \sqrt{3}\) 
   Combine the terms: 
   \(\tan A + \cot A = \frac{1 + 3}{\sqrt{3}} = \frac{4}{\sqrt{3}}\)

Therefore, the value of \( \tan A + \cot A \) is \( \frac{4}{\sqrt{3}} \), which corresponds to the correct answer option given.