Question

\[ \tan 9^\circ-\tan 27^\circ-\tan 63^\circ+\tan 81^\circ= \]

• A. \(2\)
• B. \(1\)
• C. \(4\)
• D. \(3\)

Hint:

For angles like \(9^\circ\) and \(81^\circ\), or \(27^\circ\) and \(63^\circ\), use complementary angle identities.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to evaluate a numerical trigonometric expression. Notice the relationship between the angles: \( 9 + 81 = 90 \) and \( 27 + 63 = 90 \). These are complementary angles.
Step 2: Key Formula or Approach:
1. Use complementary angle identities: \( \tan(90 - \theta) = \cot \theta \).
2. Use the identity \( \tan \theta + \cot \theta = \frac{2}{\sin 2\theta} \).
Step 3: Detailed Explanation:

Group the terms:
\( (\tan 9^\circ + \tan 81^\circ) - (\tan 27^\circ + \tan 63^\circ) \)

Apply complementarity:
\( \tan 81^\circ = \cot 9^\circ \) and \( \tan 63^\circ = \cot 27^\circ \)

The expression becomes:
\( (\tan 9^\circ + \cot 9^\circ) - (\tan 27^\circ + \cot 27^\circ) \)

Using \( \tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} = \frac{2}{\sin 2\theta} \):
\[ \frac{2}{\sin 18^\circ} - \frac{2}{\sin 54^\circ} \]

We know specific values: \( \sin 18^\circ = \frac{\sqrt{5}-1}{4} \) and \( \sin 54^\circ = \cos 36^\circ = \frac{\sqrt{5}+1}{4} \).

Substitute these:
\[ \frac{2}{(\sqrt{5}-1)/4} - \frac{2}{(\sqrt{5}+1)/4} = \frac{8}{\sqrt{5}-1} - \frac{8}{\sqrt{5}+1} \]

Take 8 as common and rationalize:
\[ 8 \left( \frac{(\sqrt{5}+1) - (\sqrt{5}-1)}{(\sqrt{5}-1)(\sqrt{5}+1)} \right) = 8 \left( \frac{2}{5-1} \right) = 8 \left( \frac{2}{4} \right) = 4 \]

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