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)

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