化简求值：$\left(1+\cos\dfrac{\pi}5\right)\left(1+\cos\dfrac{3\pi}{5}\right)=$ ．
【难度】
【出处】
无
【标注】
【答案】
$\dfrac54$
【解析】
记所求表达式为 $M$，则$$\begin{split} M&=1+\cos\dfrac{\pi}5+\cos\dfrac{3\pi}{5}+\cos\dfrac{\pi}{5}\cos\dfrac{3\pi}{5}\\
&=1+2\cos\dfrac{\pi}{5}\cos\dfrac{3\pi}{5}-\cos\dfrac{\pi}{5}\cos\dfrac{3\pi}{5}\\
&=1+\dfrac1{\sin\frac{\pi}5}\cdot\sin\dfrac{\pi}{5}\cos\dfrac{\pi}{5}\cos\dfrac{3\pi}{5}\\
&=1+\dfrac14\\
&=\dfrac54.\end{split}$$
&=1+2\cos\dfrac{\pi}{5}\cos\dfrac{3\pi}{5}-\cos\dfrac{\pi}{5}\cos\dfrac{3\pi}{5}\\
&=1+\dfrac1{\sin\frac{\pi}5}\cdot\sin\dfrac{\pi}{5}\cos\dfrac{\pi}{5}\cos\dfrac{3\pi}{5}\\
&=1+\dfrac14\\
&=\dfrac54.\end{split}$$
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