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Simplify the expression by rewriting and using identities:
We can start with the Pythagorean identity.
Now we can simplify by substituting 1+cot2θ for csc2θ. We have
Use algebraic techniques to verify the identity: cos θ1+sin θ=1−sin θcos θ.
(Hint: Multiply the numerator and denominator on the left side by 1−sin θ.)
Access these online resources for additional instruction and practice with the fundamental trigonometric identities.
Pythagorean identities | cos2θ+sin2θ=11+cot2θ=csc2θ1+tan2θ=sec2θ |
Even-odd identities | tan(−θ)=−tan θcot(−θ)=−cot θsin(−θ)=−sin θcsc(−θ)=−csc θcos(−θ)=cos θsec(−θ)=sec θ |
Reciprocal identities | sin θ=1csc θcos θ=1sec θtan θ=1cot θcsc θ=1sin θsec θ=1cos θcot θ=1tan θ |
Quotient identities | tan θ=sin θcos θcot θ=cos θsin θ |
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)=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)=(−sin x)(−sin x)=sin2x=F(x),G(−x)=cos(−x)cos(−x)=cos xcos x=cos2x=G(x) and H(−x)=tan(−x)tan(−x)=(−tan x)(−tan x)=tan2x=H(x).
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?
After examining the reciprocal identity for sec t, explain why the function is undefined at certain points.
When cos t=0, then sec t=10, which is undefined.
All of the Pythagorean identities are related. Describe how to manipulate the equations to get from sin2t+cos2t=1 to the other forms.
For the following exercises, use the fundamental identities to fully simplify the expression.
sin(−x) cos(−x) csc(−x)
csc x+cos x cot(−x)
3 sin3 t csc t+cos2 t+2 cos(−t)cos t
−sin(−x)cos x sec x csc x tan xcot x
(tan xcsc2x+tan xsec2x)(1+tan x1+cot x)−1cos2x
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.
tan x+cot xcsc x; cos x
cos x1+sin x+tan x; cos x
11−cos x−cos x1+cos x; csc x
1csc x−sin x; sec x and tan x
tan x; sec x
sec x; sin x
cot x; csc x
For the following exercises, verify the identity.
cos x−cos3x=cos x sin2 x
Answers will vary. Sample proof:
cos x−cos3x=cos x(1−cos2x)=cos xsin2x
cos x(tan x−sec(−x))=sin x−1
1+sin2xcos2x=1cos2x+sin2xcos2x=1+2 tan2x
Answers will vary. Sample proof:
1+sin2xcos2x=1cos2x+sin2xcos2x=sec2x+tan2x=tan2x+1+tan2x=1+2tan2x
(sin x+cos x)2=1+2 sin xcos x
cos2x−tan2x=2−sin2x−sec2x
Answers will vary. Sample proof:
cos2x−tan2x=1−sin2x−(sec2x−1)=1−sin2x−sec2x+1=2−sin2x−sec2x
For the following exercises, prove or disprove the identity.
11+cos x−11−cos(−x)=−2 cot x csc x
(sec2(−x)−tan2xtan x)(2+2 tan x2+2 cot x)−2 sin2x=cos 2x
sec(−x)tan x+cot x=−sin(−x)
1+sin xcos x=cos x1+sin(−x)
Proved with negative and Pythagorean identities
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.
cos2θ−sin2θ1−tan2θ=sin2θ
3 sin2θ+4 cos2θ=3+cos2θ
True 3 sin2θ+4 cos2θ=3 sin2θ+3 cos2θ+cos2θ=3(sin2θ+cos2θ)+cos2θ=3+cos2θ
sec θ+tan θcot θ+cos θ=sec2θ
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