第 4 題
若 \(\displaystyle\left(4\cos^2 9^\circ-3\right)\left(4\cos^2 27^\circ - 3\right) = \tan x^\circ\),求最小的正整數 \(x=\)?
解答:
\(\displaystyle\left(4\cos^2 9^\circ-3\right)\left(4\cos^2 27^\circ - 3\right)\)
\(\displaystyle= \frac{\left(4\cos^3 9^\circ-3\cos9^\circ\right)\left(4\cos^3 27^\circ - 3\cos 27^\circ\right)}{\cos 9^\circ \cos 27^\circ}\)
\(\displaystyle=\frac{\cos27^\circ \cos81^\circ}{\cos 9^\circ \cos 27^\circ}\)
\(\displaystyle=\frac{\cos27^\circ \sin9^\circ}{\cos 9^\circ \cos 27^\circ}\)
\(\displaystyle=\tan9^\circ.\)