\cos β_{1} - \cos β_{2} = 2 \sin \frac{β_{1} + β_{2}}{2} \sin \frac{β_{1} - β_{2}}{2}

${\cos \beta_1 - \cos \beta_2 = 2 \sin\frac{\beta_1 + \beta_2}{2}\sin \frac{\beta_1 - \beta_2}{2}}$

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