$\left(1+\cos\dfrac{\pi}5\right)\left(1+\cos\dfrac{3\pi}5\right)$ 的值为 \((\qquad)\)
【难度】
【出处】
2017年北京大学自主招生数学试题
【标注】
【答案】
B
【解析】
根据题意，有\[\begin{split}\left(1+\cos\dfrac{\pi}5\right)\left(1+\cos\dfrac{3\pi}5\right)&=1+\cos\dfrac{\pi}5+\cos\dfrac{3\pi}5+\cos\dfrac{\pi}5\cdot \cos\dfrac{3\pi}5\\
&=1+2\cos\dfrac{2\pi}5\cdot\cos\dfrac{\pi}5+\cos\dfrac{\pi}5\cdot \cos\dfrac{3\pi}5\\
&=1+\cos\dfrac{\pi}5\cdot\cos\dfrac{2\pi}5\\
&=1+\dfrac{4\sin\dfrac{\pi}5\cdot \cos\dfrac{\pi}5\cdot\cos\dfrac{2\pi}5}{4\sin\dfrac{\pi}5}\\
&=1+\dfrac 14.\end{split}\]
&=1+2\cos\dfrac{2\pi}5\cdot\cos\dfrac{\pi}5+\cos\dfrac{\pi}5\cdot \cos\dfrac{3\pi}5\\
&=1+\cos\dfrac{\pi}5\cdot\cos\dfrac{2\pi}5\\
&=1+\dfrac{4\sin\dfrac{\pi}5\cdot \cos\dfrac{\pi}5\cdot\cos\dfrac{2\pi}5}{4\sin\dfrac{\pi}5}\\
&=1+\dfrac 14.\end{split}\]
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