\(
\because \sin(\alpha+\beta)=|CD|/|OC|,|OC|=1
\)
\(
\therefore \sin(\alpha+\beta)=|CD|
\)
\(
\because \angle OCE=\pi/2-\beta,\angle OCD=\pi/2-(\beta+\alpha)
\)
\(
\therefore \angle DCE=\alpha
\)
\(
\because |CD|=|CG|+|GD|=|CG|+|EF|
\)
\(
\therefore \sin(\alpha+\beta)=|CE|\cdot\cos\alpha+|OE|\cdot\sin\alpha
\)
\(
\therefore \sin(\alpha+\beta)=|OC|\cdot\sin\beta\cdot\cos\alpha+|OC|\cdot\cos\beta\cdot\sin\alpha
\)
\(
\therefore \sin(\alpha+\beta)=\sin\beta\cdot\cos\alpha+\cos\beta\cdot\sin\alpha
\)
类似的
\(
\because \cos(\alpha+\beta)=|OD|=|OF|-|GE|
\)
\(
\therefore \cos(\alpha+\beta)=|OE|\cdot\cos\alpha-|CE|\cdot\sin\alpha
\)
\(
\therefore \cos(\alpha+\beta)=\cos\beta\cdot\cos\alpha-\sin\beta\cdot\sin\alpha
\)