Question

The value of \[ \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} will be: \]

• A. \( \frac{\sqrt{3}}{2} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{2}{\sqrt{3}} \)
• D. \( \sqrt{3} \)

Hint:

The given trigonometric identity \( \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \) is equivalent to \( \cos 2\theta \).

Answer 1

The Correct Option is B

We are given the expression: \[ \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} \] We know from standard trigonometric values that \( \tan 30^\circ = \frac{1}{\sqrt{3}} \). Now, substituting \( \tan 30^\circ = \frac{1}{\sqrt{3}} \) into the expression: \[ \frac{1 - \left(\frac{1}{\sqrt{3}}\right)^2}{1 + \left(\frac{1}{\sqrt{3}}\right)^2} = \frac{1 - \frac{1}{3}}{1 + \frac{1}{3}} = \frac{\frac{2}{3}}{\frac{4}{3}} = \frac{2}{4} = \frac{1}{2} \] Therefore, the correct answer is (B) \( \frac{1}{2} \).