Question

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

• A. 4
• B. $-4$
• C. 0
• D. 2

Answer 1

Answer: (A)

$\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$
$=\left(\tan 9^{\circ}+\tan 81^{\circ}\right)-\left(\tan 27^{\circ}+\tan 63^{\circ}\right)$
$=\frac{\sin 90^{\circ}}{\cos 9^{\circ} \cos 81^{\circ}}-\frac{\sin 90^{\circ}}{\cos 27^{\circ} \cos 63^{\circ}}=\frac{2}{2 \sin 9^{\circ} \cos 9^{\circ}}-\frac{2}{2 \sin 27^{\circ} \cos 27^{\circ}}$
$=\frac{2}{\sin 18^{\circ}}-\frac{2}{\sin 54^{\circ}}=\frac{2}{\frac{\sqrt{5}-1}{4}}-\frac{2}{\frac{\sqrt{5}+1}{4}}=\frac{8(\sqrt{5}+1-\sqrt{5}+1)}{4}=4$