 # Product to Sum Formula

Product to sum formulas are the trigonometric identities. These identities are used to rewrite products of sine and cosine. Product to sum formulas are also used to simplify the critical trigonometry function. To solve the trigonometry functions use below given Product to sum formula.

Sum to product formulas are:

\begin{array}{l}\begin{aligned} \cos (a) \cos (b) &=\frac{1}{2}(\cos (a+b)+\cos (a-b)) \\ \sin (a) \sin (b) &=\frac{1}{2}(\cos (a-b)-\cos (a+b)) \\ \sin (a) \cos (b) &=\frac{1}{2}(\sin (a+b)+\sin (a-b)) \\ \cos (a) \sin (b) &=\frac{1}{2}(\sin (a+b)-\sin (a-b)) \end{aligned}\end{array}

### Solved Example

Question: Simplify the function cos (3x) sin (2x) using product to sum formula.

Solution:

Given,

cos(3x) sin(2x)

Using formula,

cos a sin b =

$$\begin{array}{l}\frac{1}{2}\end{array}$$
$$\begin{array}{l}(sin(a + b) – sin(a – b))\end{array}$$

cos 3x sin 2x =

$$\begin{array}{l}\frac{1}{2}\end{array}$$
$$\begin{array}{l}(sin(3x + 2x) – sin(3x – 2x))\end{array}$$

cos 3x sin 2x =

$$\begin{array}{l}\frac{1}{2}\end{array}$$
$$\begin{array}{l}(sin(5x) – sin(x))\end{array}$$