Evaluate tan 9° – tan 27° – tan 63° + tan 81°

Penyelesaian soal ini sebenarnya mudah jika tahu perbandingan trigonometri untuk sudut 18°.

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

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