Download Proofs of Trigonometric Identities

Proofs of Trigonometric
Identities
Bradley Hughes
Larry Ottman
Lori Jordan
Mara Landers
Andrea Hayes
Brenda Meery
Art Fortgang
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Printed: August 21, 2013
AUTHORS
Bradley Hughes
Larry Ottman
Lori Jordan
Mara Landers
Andrea Hayes
Brenda Meery
Art Fortgang
www.ck12.org
C ONCEPT
Concept 1. Proofs of Trigonometric Identities
1
Proofs of Trigonometric
Identities
Here you’ll learn four different methods to use in proving trig identities to be true.
What if your instructor gave you two trigonometric expressions and asked you to prove that they were true. Could
you do this? For example, can you show that
sin2 θ =
1−cos 2θ
2
Read on, and in this Concept you’ll learn four different methods to help you prove identities. You’ll be able to apply
them to prove the above identity when you are finished.
Watch This
MEDIA
Click image to the left for more content.
Educator.comTrigonometric Identities
Guidance
In Trigonometry you will see complex trigonometric expressions. Often, complex trigonometric expressions can
be equivalent to less complex expressions. The process for showing two trigonometric expressions to be equivalent
(regardless of the value of the angle) is known as validating or proving trigonometric identities.
There are several options a student can use when proving a trigonometric identity.
Option One: Often one of the steps for proving identities is to change each term into their sine and cosine equivalents.
Option Two: Use the Trigonometric Pythagorean Theorem and other Fundamental Identities.
Option Three: When working with identities where there are fractions- combine using algebraic techniques for
adding expressions with unlike denominators.
Option Four: If possible, factor trigonometric expressions. For example,
2(1+cos θ)
sin θ(1+cos θ)
2+2 cos θ
sin θ(1+cos θ)
= 2 csc θ can be factored to
= 2 csc θ and in this situation, the factors cancel each other.
Example A
Prove the identity: csc θ × tan θ = sec θ
Solution: Reducing each side separately. It might be helpful to put a line down, through the equals sign. Because
we are proving this identity, we don’t know if the two sides are equal, so wait until the end to include the equality.
1
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csc x × tan x sec x
sin x
1
1
sin x × cos x
cos x
1
1
sin x
×
sinx cos x cos x
1
cos x
1
cos x
At the end we ended up with the same thing, so we know that this is a valid identity.
Notice when working with identities, unlike equations, conversions and mathematical operations are performed
only on one side of the identity. In more complex identities sometimes both sides of the identity are simplified or
expanded. The thought process for establishing identities is to view each side of the identity separately, and at the
end to show that both sides do in fact transform into identical mathematical statements.
Example B
Prove the identity: (1 − cos2 x)(1 + cot2 x) = 1
Solution: Use the Pythagorean Identity and its alternate form. Manipulate sin2 θ + cos2 θ = 1 to be sin2 θ = 1 −
cos2 θ. Also substitute csc2 x for 1 + cot2 x, then cross-cancel.
(1 − cos2 x)(1 + cot2 x)
sin2 x · csc2 x
sin2 x · sin12 x
1
1
1
1
1
Example C
Prove the identity:
sin θ
1+cos θ
θ
+ 1+cos
sin θ = 2 csc θ.
Solution: Combine the two fractions on the left side of the equation by finding the common denominator: (1 +
cos θ) × sin θ, and the change the right side into terms of sine.
sin θ
sin θ
·
sin θ
1+cos θ
1+cos θ + sin θ
sin θ
1+cos θ 1+cos θ
1+cos θ + sin θ · 1+cos θ
sin2 θ+(1+cos θ)2
sin θ(1+cos θ)
2 csc θ
2 csc θ
2 csc θ
Now, we need to apply another algebraic technique, FOIL. (FOIL is a memory device that describes the process for
multiplying two binomials, meaning multiplying the First two terms, the Outer two terms, the Inner two terms, and
then the Last two terms, and then summing the four products.) Always leave the denominator factored, because you
might be able to cancel something out at the end.
sin2 θ+1+2 cos θ+cos2 θ
sin θ(1+cos θ)
2 csc θ
Using the second option, substitute sin2 θ + cos2 θ = 1 and simplify.
2
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Concept 1. Proofs of Trigonometric Identities
1+1+2 cos θ
sin θ(1+cos θ)
2+2 cos θ
sin θ(1+cos θ)
2(1+cos θ)
sin θ(1+cos θ)
2
sin θ
2 csc θ
2 csc θ
2 csc θ
2
sin θ
Option Four: If possible, factor trigonometric expressions. Actually procedure four was used in the above example:
2(1+cos θ)
2+2 cos θ
sin θ(1+cos θ) = 2 csc θ can be factored to sin θ(1+cos θ) = 2 csc θ and in this situation, the factors cancel each other.
Vocabulary
FOIL: FOIL is a memory device that describes the process for multiplying two binomials, meaning multiplying the
First two terms, the Outer two terms, the Inner two terms, and then the Last two terms, and then summing the four
products.
Trigonometric Identity: A trigonometric identity is an expression which relates one trig function on the left side
of an equals sign to another trig function on the right side of the equals sign.
Guided Practice
1. Prove the identity: sin x tan x + cos x = sec x
2. Prove the identity: cos x − cos x sin2 x = cos3 x
3. Prove the identity:
sin x
1+cos x
x
+ 1+cos
sin x = 2 csc x
Solutions:
1. Step 1: Change everything into sine and cosine
sin x tan x + cos x = sec x
sin x
1
sin x ·
+ cos x =
cos x
cos x
Step 2: Give everything a common denominator, cos x.
sin2 x cos2 x
1
+
=
cos x
cos x
cos x
Step 3: Because the denominators are all the same, we can eliminate them.
sin2 x + cos2 x = 1
We know this is true because it is the Trig Pythagorean Theorem
2. Step 1: Pull out a cos x
cos x − cos x sin2 x = cos3 x
cos x(1 − sin2 x) = cos3 x
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Step 2: We know sin2 x + cos2 x = 1, so cos2 x = 1 − sin2 x is also true, therefore cos x(cos2 x) = cos3 x. This, of
course, is true, we are finished!
3. Step 1: Change everything in to sine and cosine and find a common denominator for left hand side.
sin x
1 + cos x
+
= 2 csc x
1 + cos x
sin x
1 + cos x
2
sin x
+
=
← LCD : sin x(1 + cos x)
1 + cos x
sin x
sin x
sin2 x + (1 + cos x)2
sin x(1 + cos x)
Step 2: Working with the left side, FOIL and simplify.
sin2 x + 1 + 2 cos x + cos2 x
sin x(1 + cos x)
sin2 x + cos2 x + 1 + 2 cos x
sin x(1 + cos x)
1 + 1 + 2 cos x
sin x(1 + cos x)
2 + 2 cos x
sin x(1 + cos x)
2(1 + cos x)
sin x(1 + cos x)
2
sin x
}}
Concept Problem Solution
The original question was to prove that:
sin2 θ =
1−cos 2θ
2
First remember the Pythagorean Identity:
sin2 θ + cos2 θ = 1
Therefore,
sin2 θ = 1 − cos2 θ
From the Double Angle Identities, we know that
cos 2θ = cos2 θ − sin2 θ
cos2 θ = cos 2θ + sin2 θ
Substituting this into the above equation for sin2 ,
4
→ FOIL (1 + cos x)2
→ move cos2 x
→ sin2 x + cos2 x = 1
→ add
→ fator out 2
→ cancel (1 + cos x)
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Concept 1. Proofs of Trigonometric Identities
sin2 θ = 1 − (cos 2θ + sin2 θ)
sin2 θ = 1 − cos 2θ − sin2 θ
2 sin2 θ = 1 − cos 2θ
1 − cos 2θ
sin2 θ =
2
Practice
Use trigonometric identities to simplify each expression as much as possible.
1.
2.
3.
4.
5.
6.
7.
8.
tan(x) cos(x)
cos(x) − cos3 (x)
1−cos2 (x)
sin(x)
cot(x) sin(x)
1−sin2 (x)
cos(x)
sin(x) csc(x)
tan(−x) cot(x)
sec2 (x)−tan2 (x)
cos2 (x)+sin2 (x)
Prove each identity.
9. tan(x) + cot(x) = sec(x) csc(x)
2
2 (x)
10. sin(x) = sin (x)+cos
csc(x)
1
1
+ sec(x)+1
= 2 cot(x) csc(x)
11. sec(x)−1
12. (cos(x))(tan(x) + sin(x) cot(x)) = sin(x) + cos2 (x)
13. sin4 (x) − cos4 (x) = sin2 (x) − cos2 (x)
14. sin2 (x) cos3 (x) = (sin2 (x) − sin4 (x))(cos(x))
sin(x)
15. csc(x)
= 1 − cos(x)
sec(x)
5