$sin(\alpha+\beta)$的证明

Output

$$\begin{array}{rl}& AB\ast AC\ast sin(\alpha +\beta )=BD\ast AD+CD\ast AD\\ \because & BD=AB\ast sin\alpha \\ & CD=AC\ast sin\beta \\ \because & AD=AB\ast cos\alpha =AC\ast cos\beta \\ & AB\ast AC\ast sin(\alpha +\beta )=AB\ast sin\alpha \ast AC\ast cos\beta \\ & +AC\ast sin\beta \ast AB\ast cos\alpha .\\ \therefore & sin(\alpha +\beta )=sin\alpha cos\beta +cos\alpha sin\beta \end{array}$$