Question

Evaluate \( \dfrac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} \)

• A. \( \sin 60^\circ \)
• B. \( \tan 60^\circ \)
• C. \( \sin 30^\circ \)
• D. \( \cot 60^\circ \)

Hint:

When evaluating trigonometric expressions, always consider simplifying directly by substituting known values of standard angles.

Answer 1

The Correct Option is A

Step 1: Use identity: \[ \tan(2A) = \frac{2 \tan A}{1 - \tan^2 A} \quad \text{But this is different, let's evaluate directly.} \] 
Step 2: Substitute values. \[ \tan 30^\circ = \frac{1}{\sqrt{3}} \Rightarrow \tan^2 30^\circ = \frac{1}{3} \] So, \[ \frac{2 \cdot \frac{1}{\sqrt{3}}}{1 + \frac{1}{3}} = \frac{\frac{2}{\sqrt{3}}}{\frac{4}{3}} = \frac{2}{\sqrt{3}} \cdot \frac{3}{4} = \frac{6}{4\sqrt{3}} = \frac{3}{2\sqrt{3}} \] Now, \[ \sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \Rightarrow \frac{3}{2\sqrt{3}} = \sin 60^\circ \]