Question

The value of \[ \tan 20^\circ \cdot \tan 80^\circ \cdot \cot 50^\circ = ? \]

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

Hint:

Use the identity \( \tan(90^\circ - x) = \cot x \) and complementary angle identities to simplify trigonometric expressions.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to simplify a product of tangent and cotangent functions. Using trigonometric identities involving triple angles is efficient.
Step 2: Key Formula or Approach:
Use the identity \( \tan \theta \tan(60^\circ - \theta) \tan(60^\circ + \theta) = \tan 3\theta \).
Step 3: Detailed Explanation:
Let \( \theta = 20^\circ \). Then \( 60^\circ - \theta = 40^\circ \) and \( 60^\circ + \theta = 80^\circ \).
The identity gives: \( \tan 20^\circ \tan 40^\circ \tan 80^\circ = \tan(3 \times 20^\circ) = \tan 60^\circ = \sqrt{3} \).
Rearranging: \( \tan 20^\circ \tan 80^\circ = \frac{\sqrt{3}}{\tan 40^\circ} = \sqrt{3} \cot 40^\circ \).
The given expression is \( (\tan 20^\circ \tan 80^\circ) \cot 50^\circ \).
Substitute: \( \sqrt{3} \cot 40^\circ \cot 50^\circ \).
Since \( \cot 50^\circ = \tan(90^\circ - 50^\circ) = \tan 40^\circ \):
Value = \( \sqrt{3} \cot 40^\circ \tan 40^\circ = \sqrt{3} \times 1 = \sqrt{3} \).
Step 4: Final Answer:
The value is \( \sqrt{3} \).