# Trigonometric Proof using Compound Angle Formula

There are many areas to apply the compound angle formulas, and trigonometric proof using compound angle formula is one of them.

\begin{aligned} \require{color} \sin (x + y) &= \sin x \cos y + \sin y \cos x &\color{green} (1) \\ \sin (x – y) &= \sin x \cos y – \sin y \cos x &\color{green} (2) \\ \end{aligned} \\
We can abstract two similar formulas using these identities for Trigonometric Proof using Compound Angle Formula.

\begin{aligned} \text{Let } A &= x + y \text{ and } B = x – y \\ A + B &= 2x \\ x &= \frac{A + B}{2} \\ A – B &= 2y \\ y &= \frac{A – B}{2} \\ \sin (x + y) + \sin (x – y) &= 2\sin x \cos y &\color{green} (1) + (2) \\ \sin A + \sin B &= 2 \sin \frac{A + B}{2} \cos \frac{A – B}{2} &\color{green} (3) \\ \sin (x + y) – \sin (x – y) &= 2\sin y \cos x &\color{green} (1) – (2) \\ \sin A – \sin B &= 2 \sin \frac{A – B}{2} \cos \frac{A + B}{2} &\color{green} (4) \\ \end{aligned} \\
The following Example Question covers one of popular ways to prove trigonometric identities.
Let’s have a look at it now!

#### Worked Example

Prove $$\sin 2A + \sin 2B + \sin 2C = 4 \sin A \sin B \sin C$$, if $$A + B + C = \pi$$.