# 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:

$\large cos\;a\;cos\;b=\frac{1}{2}\left(cos\left(a+b\right)+cos\left(a-b\right )\right)$

$\large sin\;a\;sin\;b=\frac{1}{2}\left(sin\left(a-b\right)-cos\left(a+b\right )\right)$

$\large sin\;a\;sin\;b=\frac{1}{2}\left(sin\left(a+b\right)+sin\left(a-b\right)\right)$

$\large cos\;a\;sin\;b=\frac{1}{2}\left(sin\left(a+b\right)-sin\left(a-b\right)\right)$

### solved examples

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

Solution:

Given cos(3x)sin(2x)

Using formula,

cos a sin b = $\frac{1}{2}$$(sin(a + b) – sin(a – b)) cos 3x sin 2x = \frac{1}{2}$$(sin(3x + 2x) – sin(3x – 2x))$

cos 3x sin 2x = $\frac{1}{2}$$(sin(5x) – sin(x))$

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