Question

The value of $\tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ$ is:

• A. 2
• B. 1
• C. 4
• D. 3

Hint:

Remember $\tan\theta + \cot\theta = 2\csc 2\theta$. It simplifies sums of complementary tangents immediately.

Answer 1

The Correct Option is C

Step 1: Concept
Group complementary angles and convert tangents to sines and cosines or use $\tan(90 - \theta) = \cot \theta$.

Step 2: Meaning
The expression becomes $(\tan 9^\circ + \cot 9^\circ) - (\tan 27^\circ + \cot 27^\circ)$.

Step 3: Analysis
Using $\tan \theta + \cot \theta = 2/\sin 2\theta$: The expression is $2/\sin 18^\circ - 2/\sin 54^\circ$. Since $\sin 18^\circ = (\sqrt{5}-1)/4$ and $\sin 54^\circ = (\sqrt{5}+1)/4$.

Step 4: Conclusion
$2 [ 4/(\sqrt{5}-1) - 4/(\sqrt{5}+1) ] = 8 [ (\sqrt{5}+1 - \sqrt{5} + 1) / (5 - 1) ] = 8 [2/4] = 4$.
Final Answer: (C)