每日一题[1693]半角公式

已知 $\left(\tan\alpha+\dfrac1{\cos\alpha}\right)\left(\tan\beta+\dfrac1{\cos\beta}\right)=1$，则 $\alpha+\beta=$ _______．

答案    $2k\pi$($k\in\bb Z$)．

解析    根据题意，有 $$\dfrac{1+\sin\alpha}{\cos\alpha}\cdot \dfrac{1+\sin\beta}{\cos\beta}=1,$$ 考虑半角公式，设 $\alpha=\dfrac{\pi}2-x$，$\beta=\dfrac{\pi}2-y$，则 $$\dfrac{1}{\tan\dfrac x2\tan\dfrac y2}=1\iff \tan\dfrac x2=\tan\left(\dfrac{\pi}2-\dfrac y2\right),$$ 于是 $$\dfrac x2=\dfrac{\pi}2-\dfrac y2+k\pi,k\in\mathbb Z,$$ 整理得$\alpha+\beta=2k\pi$($k\in\bb Z$)．