Question

$\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}=$

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

$\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$
$=\tan 81^{\circ}+\tan 9^{\circ}-\left(\tan 63^{\circ}+\tan 27^{\circ}\right)$
$=\cot 9^{\circ}+\tan 9^{\circ}-\left(\cot 27^{\circ}+\tan 27^{\circ}\right)$
$=\frac{\cos ^{2} 9^{\circ}+\sin ^{2} 9^{\circ}}{\sin 9^{\circ} \cdot \cos 9^{\circ}}-\frac{\cos ^{2} 27^{\circ}+\sin ^{2} 27^{\circ}}{\sin 27 \cdot \cos 27}$
$=\frac{1}{\sin 9^{\circ} \cdot \cos 9^{\circ}}-\frac{1}{\sin 27^{\circ} \cdot \cos 27^{\circ}}$
$=\frac{2}{\sin 18^{\circ}}-\frac{2}{\sin 54^{\circ}}=2 \frac{\sin 54-\sin 18}{\sin 54 \cdot \sin 18}$
$=\frac{4 \cos 36 \cdot \sin 18}{\cos 36 \sin 18}=4$