<< 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 )

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 ) .

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

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

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 ) .

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 = 1 0 , which is undefined.

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.

Algebraic

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

sin x cos x sec x

sin x

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

tan x sin x + sec x cos 2 x

sec x

csc x + cos x cot ( x )

cot t + tan t sec ( t )

csc t

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

tan ( x ) cot ( x )

−1

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

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

sec 2 x

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

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

sin 2 x + 1

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

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

1 sin x

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

1 sin x cos x cot x ; cot x

1 cot x

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

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

tan x

1 csc x sin x ; sec x  and  tan x

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

4 sec x tan x

tan x ; sec x

sec x ; cot x

± 1 cot 2 x + 1

sec x ; sin x

cot x ; sin x

± 1 sin 2 x sin x

cot x ; csc x

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

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

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

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

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

Extensions

For the following exercises, prove or disprove the identity.

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

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

False

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

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

False

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

1 + sin x cos x = cos x 1 + 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.

cos 2 θ sin 2 θ 1 tan 2 θ = sin 2 θ

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

sec θ + tan θ cot θ + cos θ = sec 2 θ

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Essential precalculus, part 2' conversation and receive update notifications?

Ask