<< Chapter < Page Chapter >> Page >

Rewriting a trigonometric expression using the difference of squares

Rewrite the trigonometric expression: 4 cos 2 θ 1.

Notice that both the coefficient and the trigonometric expression in the first term are squared, and the square of the number 1 is 1. This is the difference of squares. Thus,

4 cos 2 θ 1 = ( 2 cos θ ) 2 1                    = ( 2 cos θ 1 ) ( 2 cos θ + 1 )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Rewrite the trigonometric expression: 25 9 sin 2 θ .

This is a difference of squares formula: 25 9 sin 2 θ = ( 5 3 sin θ ) ( 5 + 3 sin θ ) .

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with the fundamental trigonometric identities.

Key equations

Pythagorean identities sin 2 θ + cos 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 by simplifying an expression and then verifying 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 ) .

Got questions? Get instant answers now!

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 ?

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Algebraic

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

sin x cos x sec x

sin x

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

tan x sin x + sec x cos 2 x

sec x

Got questions? Get instant answers now!

csc x + cos x cot ( x )

Got questions? Get instant answers now!

cot t + tan t sec ( t )

csc t

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

tan ( x ) cot ( x )

−1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

sec 2 x

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

sin 2 x + 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

1 sin x

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

1 sin x cos x cot x ; cot x

1 cot x

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

tan x

Got questions? Get instant answers now!

1 csc x sin x ; sec x  and  tan x

Got questions? Get instant answers now!

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

4 sec x tan x

Got questions? Get instant answers now!

tan x ; sec x

Got questions? Get instant answers now!

sec x ; cot x

± 1 cot 2 x + 1

Got questions? Get instant answers now!

sec x ; sin x

Got questions? Get instant answers now!

cot x ; sin x

± 1 sin 2 x sin x

Got questions? Get instant answers now!

cot x ; csc x

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Extensions

For the following exercises, prove or disprove the identity.

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

Got questions? Get instant answers now!

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

False

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

False

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Proved with negative and Pythagorean identities

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

sec θ + tan θ cot θ + cos θ = sec 2 θ

Got questions? Get instant answers now!

Questions & Answers

what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask