# 9.4 Sum-to-product and product-to-sum formulas  (Page 3/6)

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## Verifying the identity using double-angle formulas and reciprocal identities

Verify the identity $\text{\hspace{0.17em}}{\mathrm{csc}}^{2}\theta -2=\frac{\mathrm{cos}\left(2\theta \right)}{{\mathrm{sin}}^{2}\theta }.$

For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.

$\begin{array}{ccc}\hfill \frac{\mathrm{cos}\left(2\theta \right)}{{\mathrm{sin}}^{2}\theta }& =& \frac{1-2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\theta }{{\mathrm{sin}}^{2}\theta }\hfill \\ & =& \frac{1}{{\mathrm{sin}}^{2}\theta }-\frac{2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\theta }{{\mathrm{sin}}^{2}\theta }\hfill \\ & =& {\mathrm{csc}}^{2}\theta -2\hfill \end{array}$

Verify the identity $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\theta -{\mathrm{cos}}^{2}\theta ={\mathrm{sin}}^{2}\theta .$

$\begin{array}{ccc}\hfill \mathrm{tan}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\theta -{\mathrm{cos}}^{2}\theta & =& \left(\frac{\mathrm{sin}\text{\hspace{0.17em}}\theta }{\mathrm{cos}\text{\hspace{0.17em}}\theta }\right)\left(\frac{\mathrm{cos}\text{\hspace{0.17em}}\theta }{\mathrm{sin}\text{\hspace{0.17em}}\theta }\right)-{\mathrm{cos}}^{2}\theta \hfill \\ & =& 1-{\mathrm{cos}}^{2}\theta \hfill \\ & =& {\mathrm{sin}}^{2}\theta \hfill \end{array}$

Access these online resources for additional instruction and practice with the product-to-sum and sum-to-product identities.

## Key equations

 Product-to-sum Formulas $\begin{array}{ccc}\hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)+\mathrm{cos}\left(\alpha +\beta \right)\right]\hfill \\ \hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)\right]\hfill \\ \hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)-\mathrm{cos}\left(\alpha +\beta \right)\right]\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)-\mathrm{sin}\left(\alpha -\beta \right)\right]\hfill \end{array}$ Sum-to-product Formulas $\begin{array}{ccc}\hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha +\mathrm{sin}\text{\hspace{0.17em}}\beta & =& 2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\alpha +\beta }{2}\right)\mathrm{cos}\left(\frac{\alpha -\beta }{2}\right)\hfill \\ \hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha -\mathrm{sin}\text{\hspace{0.17em}}\beta & =& 2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\alpha -\beta }{2}\right)\mathrm{cos}\left(\frac{\alpha +\beta }{2}\right)\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha -\mathrm{cos}\text{\hspace{0.17em}}\beta & =& -2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\alpha +\beta }{2}\right)\mathrm{sin}\left(\frac{\alpha -\beta }{2}\right)\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha +\mathrm{cos}\text{\hspace{0.17em}}\beta & =& 2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\alpha +\beta }{2}\right)\mathrm{cos}\left(\frac{\alpha -\beta }{2}\right)\hfill \end{array}$

## Key concepts

• From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.
• We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. See [link] , [link] , and [link] .
• We can also derive the sum-to-product identities from the product-to-sum identities using substitution.
• We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See [link] .
• Trigonometric expressions are often simpler to evaluate using the formulas. See [link] .
• The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side. See [link] and [link] .

## Verbal

Starting with the product to sum formula $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =\frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)\right],$ explain how to determine the formula for $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta .$

Substitute $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ into cosine and $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ into sine and evaluate.

Provide two different methods of calculating $\text{\hspace{0.17em}}\mathrm{cos}\left(195°\right)\mathrm{cos}\left(105°\right),$ one of which uses the product to sum. Which method is easier?

Describe a situation where we would convert an equation from a sum to a product and give an example.

Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example: $\text{\hspace{0.17em}}\frac{\mathrm{sin}\left(3x\right)+\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}=1.\text{\hspace{0.17em}}$ When converting the numerator to a product the equation becomes: $\text{\hspace{0.17em}}\frac{2\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)\mathrm{cos}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}=1$

Describe a situation where we would convert an equation from a product to a sum, and give an example.

## Algebraic

For the following exercises, rewrite the product as a sum or difference.

$16\text{\hspace{0.17em}}\mathrm{sin}\left(16x\right)\mathrm{sin}\left(11x\right)$

$8\left(\mathrm{cos}\left(5x\right)-\mathrm{cos}\left(27x\right)\right)$

$20\text{\hspace{0.17em}}\mathrm{cos}\left(36t\right)\mathrm{cos}\left(6t\right)$

$2\text{\hspace{0.17em}}\mathrm{sin}\left(5x\right)\mathrm{cos}\left(3x\right)$

$\mathrm{sin}\left(2x\right)+\mathrm{sin}\left(8x\right)$

$10\text{\hspace{0.17em}}\mathrm{cos}\left(5x\right)\mathrm{sin}\left(10x\right)$

$\mathrm{sin}\left(-x\right)\mathrm{sin}\left(5x\right)$

$\frac{1}{2}\left(\mathrm{cos}\left(6x\right)-\mathrm{cos}\left(4x\right)\right)$

$\mathrm{sin}\left(3x\right)\mathrm{cos}\left(5x\right)$

For the following exercises, rewrite the sum or difference as a product.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin