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$\alpha+\beta=2k\pi$($k\in\bb Z$)．