<< 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

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, 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