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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

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit 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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