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

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Algebraic

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

16 sin ( 16 x ) sin ( 11 x )

8 ( cos ( 5 x ) cos ( 27 x ) )

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20 cos ( 36 t ) cos ( 6 t )

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2 sin ( 5 x ) cos ( 3 x )

sin ( 2 x ) + sin ( 8 x )

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10 cos ( 5 x ) sin ( 10 x )

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sin ( x ) sin ( 5 x )

1 2 ( cos ( 6 x ) cos ( 4 x ) )

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

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
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don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
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Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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