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Simplify by rewriting and using substitution

Simplify the expression by rewriting and using identities:

csc2θcot2θ

We can start with the Pythagorean identity.

1+cot2θ=csc2θ

Now we can simplify by substituting 1+cot2θ for csc2θ. We have

csc2θcot2θ=1+cot2θcot2θ=1
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Use algebraic techniques to verify the identity: cosθ1+sinθ=1sinθcosθ.

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

cosθ1+sinθ(1sinθ1sinθ)=cosθ(1sinθ)1sin2θ=cosθ(1sinθ)cos2θ=1sinθcosθ
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Access these online resources for additional instruction and practice with the fundamental trigonometric identities.

Key equations

Pythagorean identities cos2θ+sin2θ=11+cot2θ=csc2θ1+tan2θ=sec2θ
Even-odd identities tan(θ)=tanθcot(θ)=cotθsin(θ)=sinθcsc(θ)=cscθcos(θ)=cosθsec(θ)=secθ
Reciprocal identities sinθ=1cscθcosθ=1secθtanθ=1cotθcscθ=1sinθsecθ=1cosθcotθ=1tanθ
Quotient identities tanθ=sinθcosθcotθ=cosθsinθ

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)=cosx is an even function, and f(x)=sinx and h(x)=tanx are odd functions. What about G(x)=cos2x,F(x)=sin2x, and H(x)=tan2x? 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)=(sinx)(sinx)=sin2x=F(x),G(x)=cos(x)cos(x)=cosxcosx=cos2x=G(x) and H(x)=tan(x)tan(x)=(tanx)(tanx)=tan2x=H(x).

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

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

When cost=0, then sect=10, which is undefined.

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All of the Pythagorean identities are related. Describe how to manipulate the equations to get from sin2t+cos2t=1 to the other forms.

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Algebraic

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

sinxcosxsecx

sinx

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

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tanxsinx+secxcos2x

secx

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cscx+cosxcot(x)

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cott+tantsec(t)

csct

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3sin3tcsct+cos2t+2cos(t)cost

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tan(x)cot(x)

−1

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sin(x)cosxsecxcscxtanxcotx

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1+tan2θcsc2θ+sin2θ+1sec2θ

sec2x

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(tanxcsc2x+tanxsec2x)(1+tanx1+cotx)1cos2x

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1cos2xtan2x+2sin2x

sin2x+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.

tanx+cotxcscx;cosx

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secx+cscx1+tanx;sinx

1sinx

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cosx1+sinx+tanx;cosx

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1sinxcosxcotx;cotx

1cotx

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11cosxcosx1+cosx;cscx

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(secx+cscx)(sinx+cosx)2cotx;tanx

tanx

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1cscxsinx;secx and tanx

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1sinx1+sinx1+sinx1sinx;secx and tanx

4secxtanx

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tanx;secx

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secx;cotx

±1cot2x+1

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secx;sinx

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cotx;sinx

±1sin2xsinx

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cotx;cscx

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For the following exercises, verify the identity.

cosxcos3x=cosxsin2x

Answers will vary. Sample proof:

cosxcos3x=cosx(1cos2x)=cosxsin2x

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cosx(tanxsec(x))=sinx1

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1+sin2xcos2x=1cos2x+sin2xcos2x=1+2tan2x

Answers will vary. Sample proof:
1+sin2xcos2x=1cos2x+sin2xcos2x=sec2x+tan2x=tan2x+1+tan2x=1+2tan2x

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(sinx+cosx)2=1+2sinxcosx

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cos2xtan2x=2sin2xsec2x

Answers will vary. Sample proof:
cos2xtan2x=1sin2x(sec2x1)=1sin2xsec2x+1=2sin2xsec2x

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Extensions

For the following exercises, prove or disprove the identity.

11+cosx11cos(x)=2cotxcscx

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csc2x(1+sin2x)=cot2x

False

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(sec2(x)tan2xtanx)(2+2tanx2+2cotx)2sin2x=cos2x

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tanxsecxsin(x)=cos2x

False

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sec(x)tanx+cotx=sin(x)

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1+sinxcosx=cosx1+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.

cos2θsin2θ1tan2θ=sin2θ

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3sin2θ+4cos2θ=3+cos2θ

True 3sin2θ+4cos2θ=3sin2θ+3cos2θ+cos2θ=3(sin2θ+cos2θ)+cos2θ=3+cos2θ

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secθ+tanθcotθ+cosθ=sec2θ

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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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