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已知 $\sin (α + β) = \frac{1}{3}$ , $\cos (α - β) = \frac{2}{3}$ , 求 $\frac{\sin α}{\cos β} + \frac{\cos β}{\sin α} + \frac{\cos α}{\sin β} + \frac{\sin β}{\cos α}$ .

解

进行一番强劲的计算:

\begin{aligned} 原式 & = \frac{\sin α}{\cos β} + \frac{\cos β}{\sin α} + \frac{\cos α}{\sin β} + \frac{\sin β}{\cos α} \\ = \frac{\sin α \sin β + \cos α \cos β}{\cos β \sin β} + \frac{\cos β \cos α + \sin α \sin β}{\sin α \cos α} \\ = \cos (α - β) (\frac{1}{\cos β \sin β} + \frac{1}{\sin α \cos α}) = \frac{2}{3} \cdot \frac{\sin α \cos α + \sin β \cos β}{\sin α \cos α \sin β \cos β} \\ = \frac{4 (\sin 2 α + \sin 2 β)}{3 \sin 2 α \sin 2 β} = \frac{16 \sin (α + β) \cos (α - β)}{3 [\cos 2 (α - β) - \cos 2 (α + β)]} \\ = \frac{16 \cdot \frac{1}{3} \cdot \frac{2}{3}}{3 [2 \cdot \frac{4}{9} - 1 - 1 + 2 \cdot \frac{1}{9}]} = - \frac{4}{3} . \end{aligned}