$\cos \dfrac{\pi}{9}+\cos\dfrac{3\pi}{9}+\cos\dfrac{5\pi}{9}+\cos\dfrac{7\pi}{9}=$ ．
【难度】
【出处】
无
【标注】
【答案】
$\dfrac12$
【解析】
记所求表达式为 $M$，则$$\begin{split} M&=\cos \dfrac{\pi}{9}+\cos\dfrac{7\pi}{9}+\cos\dfrac{3\pi}{9}+\cos\dfrac{5\pi}{9}\\
&=2\cos\dfrac{4\pi}{9}\left(\cos\dfrac{3\pi}{9}+\cos\dfrac{\pi}{9}\right)\\
&=4\cos\dfrac{4\pi}{9}\cos\dfrac{2\pi}{9}\cos\dfrac{\pi}{9}\\
&=\dfrac1{\sin\frac{\pi}{9}}\cdot4\cos\dfrac{4\pi}{9}\cos\dfrac{2\pi}{9}\cos\dfrac{\pi}{9}\sin\dfrac{\pi}{9}\\
&=\dfrac12.
\end{split}$$
&=2\cos\dfrac{4\pi}{9}\left(\cos\dfrac{3\pi}{9}+\cos\dfrac{\pi}{9}\right)\\
&=4\cos\dfrac{4\pi}{9}\cos\dfrac{2\pi}{9}\cos\dfrac{\pi}{9}\\
&=\dfrac1{\sin\frac{\pi}{9}}\cdot4\cos\dfrac{4\pi}{9}\cos\dfrac{2\pi}{9}\cos\dfrac{\pi}{9}\sin\dfrac{\pi}{9}\\
&=\dfrac12.
\end{split}$$
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