Question

\(\tan 10^\circ \tan 23^\circ \tan 80^\circ \tan 67^\circ = \)

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

Hint:

When you see a product of tangent functions, immediately check for pairs of angles that add up to 90°. Each such pair \(\tan A \tan B\) will simplify to 1.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves simplifying a product of tangent functions by using the complementary angle identity.

Step 2: Key Formula or Approach:
The key identities are:
1. \(\tan(90^\circ - \theta) = \cot \theta\)
2. \(\tan \theta \times \cot \theta = 1\)

Step 3: Detailed Explanation:
The given expression is \(\tan 10^\circ \tan 23^\circ \tan 80^\circ \tan 67^\circ\).
Let's pair the angles that add up to 90°: (10°, 80°) and (23°, 67°).
Now, convert one angle from each pair using the identity \(\tan(90^\circ - \theta) = \cot \theta\).
\[ \tan 80^\circ = \tan(90^\circ - 10^\circ) = \cot 10^\circ \] \[ \tan 67^\circ = \tan(90^\circ - 23^\circ) = \cot 23^\circ \] Substitute these back into the expression:
\[ \tan 10^\circ \tan 23^\circ \cot 10^\circ \cot 23^\circ \] Group the terms with the same angle:
\[ (\tan 10^\circ \cot 10^\circ) \times (\tan 23^\circ \cot 23^\circ) \] Since \(\tan \theta \cot \theta = 1\), the expression becomes:
\[ 1 \times 1 = 1 \]

Step 4: Final Answer:
The value of the expression is 1.