Proof 1 \( \sin (\alpha-\beta) = \sin \alpha \cos \beta-\cos \alpha \sin \beta \) \( \require{AMSsymbols} \begin{align} \angle RPN &= 90^{\circ}-\angle PNR \\ &= \angle RNB \\ &= \angle QON \\ &= \alpha \\ \sin(\alpha-\beta) &= \sin \angle MOP \\ &= \displaystyle \frac{MP}{OP} \\ &= \frac{MR-PR}{OP} \\ &= \frac{QN}{OP}-\frac{PR}{OP} \\ &= \frac{QN}{\color{red}{ON}} \times \frac{\color{red}{ON}}{OP}-\frac{PR}{\color{red}{PN}} \times […]
