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Simplify by rewriting and using substitution

Simplify the expression by rewriting and using identities:

csc 2 θ cot 2 θ

We can start with the Pythagorean identity.

1 + cot 2 θ = csc 2 θ

Now we can simplify by substituting 1 + cot 2 θ for csc 2 θ . We have

csc 2 θ cot 2 θ = 1 + cot 2 θ cot 2 θ = 1
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Use algebraic techniques to verify the identity: cos θ 1 + sin θ = 1 sin θ cos θ .

(Hint: Multiply the numerator and denominator on the left side by 1 sin θ . )

cos θ 1 + sin θ ( 1 sin θ 1 sin θ ) = cos θ ( 1 sin θ ) 1 sin 2 θ = cos θ ( 1 sin θ ) cos 2 θ = 1 sin θ cos θ
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Access these online resources for additional instruction and practice with the fundamental trigonometric identities.

Key equations

Pythagorean identities cos 2 θ + sin 2 θ = 1 1 + cot 2 θ = csc 2 θ 1 + tan 2 θ = sec 2 θ
Even-odd identities tan ( θ ) = tan θ cot ( θ ) = cot θ sin ( θ ) = sin θ csc ( θ ) = csc θ cos ( θ ) = cos θ sec ( θ ) = sec θ
Reciprocal identities sin θ = 1 csc θ cos θ = 1 sec θ tan θ = 1 cot θ csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ
Quotient identities tan θ = sin θ cos θ cot θ = cos θ sin θ

Key concepts

  • There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem.
  • Graphing both sides of an identity will verify it. See [link] .
  • Simplifying one side of the equation to equal the other side is another method for verifying an identity. See [link] and [link] .
  • The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation. See [link] .
  • We can create an identity and then verify it. See [link] .
  • Verifying an identity may involve algebra with the fundamental identities. See [link] and [link] .
  • Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics. See [link] , [link] , and [link] .

Section exercises

Verbal

We know g ( x ) = cos x is an even function, and f ( x ) = sin x and h ( x ) = tan x are odd functions. What about G ( x ) = cos 2 x , F ( x ) = sin 2 x , and H ( x ) = tan 2 x ? Are they even, odd, or neither? Why?

All three functions, F , G , and H , are even.

This is because F ( x ) = sin ( x ) sin ( x ) = ( sin x ) ( sin x ) = sin 2 x = F ( x ) , G ( x ) = cos ( x ) cos ( x ) = cos x cos x = cos 2 x = G ( x ) and H ( x ) = tan ( x ) tan ( x ) = ( tan x ) ( tan x ) = tan 2 x = H ( x ) .

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Examine the graph of f ( x ) = sec x on the interval [ π , π ] . How can we tell whether the function is even or odd by only observing the graph of f ( x ) = sec x ?

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After examining the reciprocal identity for sec t , explain why the function is undefined at certain points.

When cos t = 0 , then sec t = 1 0 , which is undefined.

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All of the Pythagorean identities are related. Describe how to manipulate the equations to get from sin 2 t + cos 2 t = 1 to the other forms.

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Algebraic

For the following exercises, use the fundamental identities to fully simplify the expression.

sin x cos x sec x

sin x

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sin ( x ) cos ( x ) csc ( x )

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tan x sin x + sec x cos 2 x

sec x

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csc x + cos x cot ( x )

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cot t + tan t sec ( t )

csc t

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3 sin 3 t csc t + cos 2 t + 2 cos ( t ) cos t

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tan ( x ) cot ( x )

−1

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sin ( x ) cos x sec x csc x tan x cot x

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1 + tan 2 θ csc 2 θ + sin 2 θ + 1 sec 2 θ

sec 2 x

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( tan x csc 2 x + tan x sec 2 x ) ( 1 + tan x 1 + cot x ) 1 cos 2 x

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1 cos 2 x tan 2 x + 2 sin 2 x

sin 2 x + 1

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For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

tan x + cot x csc x ; cos x

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sec x + csc x 1 + tan x ; sin x

1 sin x

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cos x 1 + sin x + tan x ; cos x

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1 sin x cos x cot x ; cot x

1 cot x

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1 1 cos x cos x 1 + cos x ; csc x

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( sec x + csc x ) ( sin x + cos x ) 2 cot x ; tan x

tan x

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1 csc x sin x ; sec x  and  tan x

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1 sin x 1 + sin x 1 + sin x 1 sin x ; sec x  and  tan x

4 sec x tan x

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tan x ; sec x

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sec x ; cot x

± 1 cot 2 x + 1

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sec x ; sin x

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cot x ; sin x

± 1 sin 2 x sin x

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cot x ; csc x

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For the following exercises, verify the identity.

cos x cos 3 x = cos x sin 2 x

Answers will vary. Sample proof:

cos x cos 3 x = cos x ( 1 cos 2 x ) = cos x sin 2 x

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cos x ( tan x sec ( x ) ) = sin x 1

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1 + sin 2 x cos 2 x = 1 cos 2 x + sin 2 x cos 2 x = 1 + 2 tan 2 x

Answers will vary. Sample proof:
1 + sin 2 x cos 2 x = 1 cos 2 x + sin 2 x cos 2 x = sec 2 x + tan 2 x = tan 2 x + 1 + tan 2 x = 1 + 2 tan 2 x

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( sin x + cos x ) 2 = 1 + 2 sin x cos x

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cos 2 x tan 2 x = 2 sin 2 x sec 2 x

Answers will vary. Sample proof:
cos 2 x tan 2 x = 1 sin 2 x ( sec 2 x 1 ) = 1 sin 2 x sec 2 x + 1 = 2 sin 2 x sec 2 x

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Extensions

For the following exercises, prove or disprove the identity.

1 1 + cos x 1 1 cos ( x ) = 2 cot x csc x

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csc 2 x ( 1 + sin 2 x ) = cot 2 x

False

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( sec 2 ( x ) tan 2 x tan x ) ( 2 + 2 tan x 2 + 2 cot x ) 2 sin 2 x = cos 2 x

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tan x sec x sin ( x ) = cos 2 x

False

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sec ( x ) tan x + cot x = sin ( x )

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1 + sin x cos x = cos x 1 + sin ( x )

Proved with negative and Pythagorean identities

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For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.

cos 2 θ sin 2 θ 1 tan 2 θ = sin 2 θ

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3 sin 2 θ + 4 cos 2 θ = 3 + cos 2 θ

True 3 sin 2 θ + 4 cos 2 θ = 3 sin 2 θ + 3 cos 2 θ + cos 2 θ = 3 ( sin 2 θ + cos 2 θ ) + cos 2 θ = 3 + cos 2 θ

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sec θ + tan θ cot θ + cos θ = sec 2 θ

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Questions & Answers

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Practice Key Terms 4

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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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