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# Sum-to-Product and Product-to-Sum Identities

Trigonometric equations can be difficult to work with, but we have a few tools at our disposal to help us out. Some of the most important are the sum-to-product and product-to-sum identities, which are kind of like trigonometric properties we can use to rewrite expressions and make them easier to work with. In this article, we'll list all of these identities and supply examples and where and when we can apply them. Let's get going!

## The sum-to-product identities

There are 4 sum-to-product identities:

1. $\mathrm{sin}\left(u\right)+\mathrm{sin}\left(v\right)=2\mathrm{sin}\left(\frac{u+v}{2}\right)\mathrm{cos}\left(\frac{u-v}{2}\right)$
2. $\mathrm{sin}\left(u\right)-\mathrm{sin}\left(v\right)=2\mathrm{cos}\left(\frac{u+v}{2}\right)\mathrm{sin}\left(\frac{u-v}{2}\right)$
3. $\mathrm{cos}\left(u\right)+\mathrm{cos}\left(v\right)=2\mathrm{cos}\left(\frac{u+v}{2}\right)\mathrm{cos}\left(\frac{u-v}{2}\right)$
4. $\mathrm{cos}\left(u\right)-\mathrm{cos}\left(v\right)=-2\mathrm{sin}\left(\frac{u+v}{2}\right)\mathrm{sin}\left(\frac{u-v}{2}\right)$

That might look like a lot, but it isn't too bad. On the left side, we have $\mathrm{sin}+\mathrm{sin}$ , $\mathrm{sin}-\mathrm{sin}$ , $\mathrm{cos}+\mathrm{cos}$ , and $\mathrm{cos}-\mathrm{cos}$ . On the right side, there's always a 2 involved in the first term and the operations inside the parenthesis is the same across all four. Therefore, we only need to memorize the sines and cosines as opposed to everything above.

Let's try applying some of these identities by expressing $\mathrm{cos}\left(6x\right)+\mathrm{cos}\left(2x\right)$ as a product. Since we're adding two cosines, we want to use the identity labeled as number 3 above. Our $u$ value is $6x$ and our $v$ value is $2x$ , allowing us to rewrite the expression like this:

$2\mathrm{cos}\left(\frac{6x+2x}{2}\right)\mathrm{cos}\left(\frac{6x-2x}{2}\right)$

We can simplify further since we have a 2 in the denominator and even numbers in the numerator:

$2\mathrm{cos}\left(4x\right)\mathrm{cos}\left(2x\right)$

We've expressed the sum as a product!

## The product-to sum identities

Above, we transformed a sum into a product. However, we can also express a product as a sum using the following three product-to-sum identities:

1. $\mathrm{sin}\left(u\right)\mathrm{sin}\left(v\right)=\frac{1}{2}\left[\mathrm{cos}\left(u-v\right)-\mathrm{cos}\left(u+v\right)\right]$

2. $\mathrm{cos}\left(u\right)\mathrm{cos}\left(v\right)=\frac{1}{2}\left[\mathrm{cos}\left(u-v\right)+\mathrm{cos}\left(u+v\right)\right]$

3. $\mathrm{sin}\left(u\right)\mathrm{cos}\left(v\right)=\frac{1}{2}\left[\mathrm{sin}\left(u-v\right)+\mathrm{sin}\left(u+v\right)\right]$

Again, this looks more difficult to memorize than it is. We have $\mathrm{sin}\left(\mathrm{sin}\right)$ , $\mathrm{cos}\left(\mathrm{cos}\right)$ , and $\mathrm{sin}\left(\mathrm{cos}\right)$ , every combination imaginable. The stuff in the brackets never changes, and everything is multiplied by $\frac{1}{2}$ . We only have to remember what's cosine and what's sine, and all three only include one or the other.

Let's try putting these identities into practice by expressing the product $\mathrm{cos}\left(3x\right)\mathrm{sin}\left(2x\right)$ as a sum of trigonometric functions. Since our product involves a sine and a cosine, we'll be using number 3 above. We get:

$\frac{1}{2}\left[\mathrm{sin}\left(3x+2x\right)-\mathrm{sin}\left(3x-2x\right)\right]$

$\frac{1}{2}\left[\mathrm{sin}\left(5x\right)-\mathrm{sin}\left(3x-2x\right)\right]$

## Practice Questions

a. Express as a product of trigonometric functions: $\mathrm{sin}\left(x\right)+\mathrm{sin}\left(2x\right)$

Since we're adding two sines, we're using $\mathrm{sin}\left(u\right)+\mathrm{sin}\left(v\right)=2\mathrm{sin}\left(\frac{u+v}{2}\right)\mathrm{cos}\left(\frac{u-v}{2}\right)$ as our identity. Our $u$ value is $x$ and our $v$ value is $2v$ giving us:

$2\mathrm{sin}\left(\frac{x+2x}{2}\right)\mathrm{cos}\left(\frac{x-2x}{2}\right)$

$2\mathrm{sin}\left(\frac{3x}{2}\right)\mathrm{cos}\left(\frac{-x}{2}\right)$

b. Express as a sum of trigonometric functions: $\mathrm{cos}\left(4x\right)\mathrm{cos}\left(3x\right)$

Since we're multiplying two cosines, we're using $\mathrm{cos}\left(u\right)\mathrm{cos}\left(v\right)=\frac{1}{2}\left[\mathrm{cos}\left(u-v\right)+\mathrm{cos}\left(u+v\right)\right]$ as our identity. Our $u$ value is $4x$ and our $v$ value is $3x$ , giving us:

$\frac{1}{2}\left[\mathrm{cos}\left(4x-3x\right)+\mathrm{cos}\left(4x+3x\right)\right]$

$\frac{1}{2}\left[\mathrm{cos}\left(x\right)+\mathrm{cos}\left(7x\right)\right]$

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There are a total of seven trigonometric sum-to-product and product-to-sum identities for students to remember, and the rote memorization involved can feel overwhelming. If your student could benefit from a helping hand, a private math tutor can share helpful mnemonics and other memory tricks to make things easier. They can also share practical applications to make the information feel less abstract. If you'd like to find out more about the benefits of one-on-one tutoring, contact Varsity Tutors today.

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