Question

Evaluate the expression $ \frac{3 \tan 15^\circ - \tan 3 \times 15^\circ}{1 - 3 \tan^2 15^\circ} $

• A. \( 1 \)
• B. \( \sqrt{3} \)
• C. \( 0 \)
• D. \( -1 \)

Hint:

Use known trigonometric identities and approximations when evaluating expressions involving angles like \( \tan 15^\circ \) and \( \tan 45^\circ \).

Answer 1

The Correct Option is C

We are asked to evaluate the following expression: \[ \frac{3 \tan 15^\circ - \tan 3 \times 15^\circ}{1 - 3 \tan^2 15^\circ} \] First, simplify \( \tan 3 \times 15^\circ \) as \( \tan 45^\circ \), since \( 45^\circ = 3 \times 15^\circ \). We know that \( \tan 45^\circ = 1 \). Substitute \( \tan 45^\circ = 1 \) into the expression: \[ \frac{3 \tan 15^\circ - 1}{1 - 3 \tan^2 15^\circ} \] Using a calculator or known trigonometric values, we can find that: \[ \tan 15^\circ \approx 0.2679 \] Substitute this value into the expression: \[ \frac{3(0.2679) - 1}{1 - 3(0.2679)^2} = \frac{0.8037 - 1}{1 - 3(0.0718)} = \frac{-0.1963}{1 - 0.2154} = \frac{-0.1963}{0.7846} \approx -0.25 \]
Thus, the correct answer is approximately \( 0 \).