Question

The value of expression \(\tan 9^{\circ }-\tan 27^{\circ }-\tan 63^{\circ }+\tan 81^{\circ }\) is....

• A. \(1\)
• B. \(2\)
• C. \(4\)
• D. \(8\)

Hint:

Group tan 9 with tan 81 and tan 27 with tan 63, then use tan x + cot x = 2 / sin 2x.

Answer 1

The Correct Option is C

Step 1: Group the Terms
Since \(\tan 81^{\circ}=\cot 9^{\circ}\) and \(\tan 63^{\circ}=\cot 27^{\circ}\), the expression is \((\tan 9^{\circ}+\cot 9^{\circ})-(\tan 27^{\circ}+\cot 27^{\circ})\).

Step 2: Use the Identity
\(\tan x+\cot x=\dfrac{2}{\sin 2x}\). The expression becomes \(\dfrac{2}{\sin 18^{\circ}}-\dfrac{2}{\sin 54^{\circ}}\).

Step 3: Insert Values
\(\sin 18^{\circ}=\dfrac{\sqrt{5}-1}{4}\) and \(\sin 54^{\circ}=\dfrac{\sqrt{5}+1}{4}\), so \(\dfrac{2}{\sin 18^{\circ}}=2(\sqrt{5}+1)\) and \(\dfrac{2}{\sin 54^{\circ}}=2(\sqrt{5}-1)\).

Step 4: Subtract
\(2(\sqrt{5}+1)-2(\sqrt{5}-1)=4\), option (C).

Final Answer:
The value of the expression is 4. \[ \boxed{\text{(C) }4} \]