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Proving an identity

Prove the identity:

cos ( 4 t ) cos ( 2 t ) sin ( 4 t ) + sin ( 2 t ) = tan t

We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.

cos ( 4 t ) cos ( 2 t ) sin ( 4 t ) + sin ( 2 t ) = 2 sin ( 4 t + 2 t 2 ) sin ( 4 t 2 t 2 ) 2 sin ( 4 t + 2 t 2 ) cos ( 4 t 2 t 2 )                             = 2 sin ( 3 t ) sin t 2 sin ( 3 t ) cos t                             = 2 sin ( 3 t ) sin t 2 sin ( 3 t ) cos t                             = sin t cos t                             = tan t
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Verifying the identity using double-angle formulas and reciprocal identities

Verify the identity csc 2 θ 2 = cos ( 2 θ ) sin 2 θ .

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.

cos ( 2 θ ) sin 2 θ = 1 2 sin 2 θ sin 2 θ              = 1 sin 2 θ 2 sin 2 θ sin 2 θ              = csc 2 θ 2
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Verify the identity tan θ cot θ cos 2 θ = sin 2 θ .

tan θ cot θ cos 2 θ = ( sin θ cos θ ) ( cos θ sin θ ) cos 2 θ = 1 cos 2 θ = sin 2 θ

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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 cos α cos β = 1 2 [ cos ( α β ) + cos ( α + β ) ] sin α cos β = 1 2 [ sin ( α + β ) + sin ( α β ) ] sin α sin β = 1 2 [ cos ( α β ) cos ( α + β ) ] cos α sin β = 1 2 [ sin ( α + β ) sin ( α β ) ]
Sum-to-product Formulas sin α + sin β = 2 sin ( α + β 2 ) cos ( α β 2 ) sin α sin β = 2 sin ( α β 2 ) cos ( α + β 2 ) cos α cos β = 2 sin ( α + β 2 ) sin ( α β 2 ) cos α + cos β = 2 cos ( α + β 2 ) cos ( α β 2 )

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] .

Section exercises

Verbal

Starting with the product to sum formula sin α cos β = 1 2 [ sin ( α + β ) + sin ( α β ) ] , explain how to determine the formula for cos α sin β .

Substitute α into cosine and β into sine and evaluate.

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Explain two different methods of calculating cos ( 195° ) cos ( 105° ) , one of which uses the product to sum. Which method is easier?

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Explain 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: sin ( 3 x ) + sin x cos x = 1. When converting the numerator to a product the equation becomes: 2 sin ( 2 x ) cos x cos x = 1

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Questions & Answers

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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