Question

tan 5° · tan 25° · tan 30° · tan 65° · tan 85° =

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

Hint:

When dealing with products of tangent and cotangent functions, especially involving complementary angles, remember to simplify before final calculations.

Answer 1

The Correct Option is C

Step 1: Use the identity \( \tan(90^\circ - \theta) = \cot(\theta) \). Thus, \( \tan 85^\circ = \cot 5^\circ \) and \( \tan 65^\circ = \cot 25^\circ \). Step 2: The expression becomes: \[ \tan 5^\circ \cdot \tan 25^\circ \cdot \tan 30^\circ \cdot \cot 25^\circ \cdot \cot 5^\circ = \tan 30^\circ \] Step 3: Simplify: \[ \tan 30^\circ = \frac{1}{\sqrt{3}} \] Thus, the correct answer is \( \boxed{\frac{1}{\sqrt{3}}} \).