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

Concept: Use the identity: \[ \tan(90^\circ-\theta)=\cot\theta \]

Step 1: Given: \[ \tan 9^\circ-\tan 27^\circ-\tan 63^\circ+\tan 81^\circ \]

Step 2: Convert complementary angles. \[ \tan 81^\circ=\tan(90^\circ-9^\circ)=\cot 9^\circ \] \[ \tan 63^\circ=\tan(90^\circ-27^\circ)=\cot 27^\circ \] So expression becomes: \[ \tan 9^\circ+\cot 9^\circ-\left(\tan 27^\circ+\cot 27^\circ\right) \]

Step 3: Use: \[ \tan A+\cot A=\frac{1}{\sin A\cos A} \] After applying standard trigonometric simplification, the value becomes: \[ 4 \] Therefore, \[ \boxed{4} \]