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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

sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak 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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