This identity can be proved by using the area method. The area method is discussed in the book Geometry Technique. \begin{align*} S_{\triangle{PAB}} &= S_{\triangle{PAC}}+S_{\triangle{PBC}}\\ \frac{1}{2}\cdot PA \cdot PB \cdot \sin(\alpha+\beta) &= \frac{1}{2}\cdot PA \cdot PC \cdot\sin\alpha + \frac{1}{2}\cdot PB \cdot PC \cdot \sin\beta \end{align*} Then dividing both sides by $\Big(\frac{1}{2}\cdot PA\cdot PB\cdot PC\Big)$ immediately leads to the result. Note: when \(PC\perp AB\), the conclusion can be used to prove the trigonometric sum of angle formula.