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: Identifying trigonometric identities.
We are given the expression:
\[ \tan 20^\circ \cdot \tan 80^\circ \cdot \cot 50^\circ. \]
We know the following trigonometric identities:
\[ \tan(90^\circ - x) = \cot x. \]
This identity tells us that:
\[ \tan 80^\circ = \cot 10^\circ. \]
Thus, we can rewrite the expression as: \[ \tan 20^\circ \cdot \cot 10^\circ \cdot \cot 50^\circ. \]

Step 2: Simplifying using complementary angles.
Next, we know that:
\[ \cot(90^\circ - x) = \tan x. \]
Using this identity, we can simplify \( \cot 50^\circ \) as:
\[ \cot 50^\circ = \tan 40^\circ. \]
Substitute this into the expression:
\[ \tan 20^\circ \cdot \tan 40^\circ \cdot \cot 50^\circ = \tan 20^\circ \cdot \tan 40^\circ. \]

Step 3: Using the tangent addition formula.
We use the following identity for the tangent of the sum of two angles:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}. \]
For \( A = 20^\circ \) and \( B = 40^\circ \), we have:
\[ \tan(20^\circ + 40^\circ) = \frac{\tan 20^\circ + \tan 40^\circ}{1 - \tan 20^\circ \cdot \tan 40^\circ} = \tan 60^\circ. \]
We know that \( \tan 60^\circ = \sqrt{3} \), so: \[ \tan 20^\circ \cdot \tan 40^\circ = \sqrt{3}. \]

Step 4: Conclusion.
The given expression simplifies to:
\[ \tan 20^\circ \cdot \tan 80^\circ \cdot \cot 50^\circ = 2 \sqrt{3}. \]
Thus, the correct answer is: \[ \boxed{2 \sqrt{3}}. \]