There are many areas to apply the compound angle formulas, and trigonometric proof using the 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!

Example

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

\( \begin{aligned} \displaystyle

\text{LHS} &= \sin 2A + \sin 2B + \sin 2C \\

&= 2 \sin (A + B) \cos (A – B) + \sin 2C &\color{green} \text{apply (3)} \\

&= 2 \sin (A + B) \cos (A – B) + 2 \sin C \cos C &\color{green} \text{double angle formula} \\

&= 2 \sin (\pi – C) \cos(A – B) + 2 \sin C \cos (\pi – A – B) &\color{green} A + B + C = \pi \\

&= 2 \sin C \cos(A – B) + 2 \sin C \cos (\pi – A – B) &\color{green} \sin (\pi – \theta) = \sin \theta \\

&= 2 \sin C \cos(A – B) – 2 \sin C \cos (A + B) &\color{green} \cos (\pi – \theta) = -\cos \theta \\

&= 2 \sin C \big[\cos (A – B) – \cos(A + B)\big] &\color{green} \text{common factor of } 2 \sin C \\

&= 2 \sin C \big[(\cos A \cos B + \sin A \sin B) – (\cos A \cos B – \sin A \sin B)\big] &\color{green} \text{compound angle formulas} \\

&= 2 \sin C (\cos A \cos B + \sin A \sin B – \cos A \cos B + \sin A \sin B) \\

&= 2 \sin C (2 \sin A \sin B) \\

&= 4 \sin C \sin A \sin B \\

&= \text{RHS} \\

\end{aligned} \\ \)

Absolute Value Algebra Arithmetic Mean Arithmetic Sequence Binomial Expansion Binomial Theorem Chain Rule Circle Geometry Common Difference Common Ratio Compound Interest Cyclic Quadrilateral Differentiation Discriminant Double-Angle Formula Equation Exponent Exponential Function Factorials Functions Geometric Mean Geometric Sequence Geometric Series Inequality Integration Integration by Parts Kinematics Logarithm Logarithmic Functions Mathematical Induction Polynomial Probability Product Rule Proof Quadratic Quotient Rule Rational Functions Sequence Sketching Graphs Surds Transformation Trigonometric Functions Trigonometric Properties VCE Mathematics Volume