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

 Page 3 / 6

## 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.

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
1+cos²A/cos²A=2cosec²A-1
test for convergence the series 1+x/2+2!/9x3
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
Ajith
exponential series
Naveen
what is subgroup
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
e power cos hyperbolic (x+iy)
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
why {2kπ} union {kπ}={kπ}?
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
what is complex numbers
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Solve 2cos x + 3sin x = 0.5