填10
原式\(\displaystyle=\frac{2}{1+\cos20^\circ}+\frac{1}{1-\cos^220^\circ}+\frac{1}{4(1-\cos^220^\circ)\cos^220^\circ}\)
\(\displaystyle=\frac{12\cos^220^\circ-8\cos^320^\circ+1}{4(1-\cos^220^\circ)\cos^220^\circ}\) \(\left(-8\cos^320^\circ+6\cos20^\circ=-1\right)\)
\(\displaystyle=\frac{12\cos^220^\circ-6\cos20^\circ}{4(1-\cos^220^\circ)\cos^220^\circ}=\frac{6\cos20^\circ-3}{2(1-\cos^220^\circ)\cos20^\circ}=\frac{6\cos20^\circ-3}{2\cos20^\circ-2\cos^320^\circ}\) \(\displaystyle\left(-2\cos^320^\circ+\frac{3}{2}\cos20^\circ=-\frac{1}{4}\right)\) 其實考試當下看到\(\displaystyle-\frac{1}{4}\)我就已經先填答案是\(12\)了
\(\displaystyle=\frac{6\cos20^\circ-3}{\displaystyle\frac{1}{2}\cos20^\circ-\frac{1}{4}}=12\)
