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Precalculus

7.1 Solving Trigonometric Equations with Identities

Precalculus7.1 Solving Trigonometric Equations with Identities
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section, you will:
  • Verify the fundamental trigonometric identities.
  • Simplify trigonometric expressions using algebra and the identities.
Photo of international passports.
Figure 1 International passports and travel documents

In espionage movies, we see international spies with multiple passports, each claiming a different identity. However, we know that each of those passports represents the same person. The trigonometric identities act in a similar manner to multiple passports—there are many ways to represent the same trigonometric expression. Just as a spy will choose an Italian passport when traveling to Italy, we choose the identity that applies to the given scenario when solving a trigonometric equation.

In this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions.

Verifying the Fundamental Trigonometric Identities

Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools of solving algebraic equations. In fact, we use algebraic techniques constantly to simplify trigonometric expressions. Basic properties and formulas of algebra, such as the difference of squares formula and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations. We already know that all of the trigonometric functions are related because they all are defined in terms of the unit circle. Consequently, any trigonometric identity can be written in many ways.

To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation. Sometimes we have to factor expressions, expand expressions, find common denominators, or use other algebraic strategies to obtain the desired result. In this first section, we will work with the fundamental identities: the Pythagorean Identities, the even-odd identities, the reciprocal identities, and the quotient identities.

We will begin with the Pythagorean Identities (see Table 1), which are equations involving trigonometric functions based on the properties of a right triangle. We have already seen and used the first of these identifies, but now we will also use additional identities.

Pythagorean Identities
sin 2 θ+ cos 2 θ=1 sin 2 θ+ cos 2 θ=1 1+ cot 2 θ= csc 2 θ 1+ cot 2 θ= csc 2 θ 1+ tan 2 θ= sec 2 θ 1+ tan 2 θ= sec 2 θ
Table 1

The second and third identities can be obtained by manipulating the first. The identity 1+ cot 2 θ= csc 2 θ 1+ cot 2 θ= csc 2 θ is found by rewriting the left side of the equation in terms of sine and cosine.

Prove: 1+ cot 2 θ= csc 2 θ 1+ cot 2 θ= csc 2 θ

1+ cot 2 θ=( 1+ cos 2 θ sin 2 θ ) Rewrite the left side.              =( sin 2 θ sin 2 θ )+( cos 2 θ sin 2 θ ) Write both terms with the common denominator.              = sin 2 θ+ cos 2 θ sin 2 θ              = 1 sin 2 θ              = csc 2 θ 1+ cot 2 θ=( 1+ cos 2 θ sin 2 θ ) Rewrite the left side.              =( sin 2 θ sin 2 θ )+( cos 2 θ sin 2 θ ) Write both terms with the common denominator.              = sin 2 θ+ cos 2 θ sin 2 θ              = 1 sin 2 θ              = csc 2 θ

Similarly, 1+ tan 2 θ= sec 2 θ 1+ tan 2 θ= sec 2 θ can be obtained by rewriting the left side of this identity in terms of sine and cosine. This gives

1+ tan 2 θ=1+ ( sinθ cosθ ) 2 Rewrite left side. = ( cosθ cosθ ) 2 + ( sinθ cosθ ) 2 Write both terms with the common denominator. = cos 2 θ+ sin 2 θ cos 2 θ = 1 cos 2 θ = sec 2 θ 1+ tan 2 θ=1+ ( sinθ cosθ ) 2 Rewrite left side. = ( cosθ cosθ ) 2 + ( sinθ cosθ ) 2 Write both terms with the common denominator. = cos 2 θ+ sin 2 θ cos 2 θ = 1 cos 2 θ = sec 2 θ

The next set of fundamental identities is the set of even-odd identities. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity is odd or even. (See Table 2).

Even-Odd Identities
tan(θ)=tanθ cot(θ)=cotθ tan(θ)=tanθ cot(θ)=cotθ sin(θ)=sinθ csc(θ)=cscθ sin(θ)=sinθ csc(θ)=cscθ cos(θ)=cosθ sec(θ)=secθ cos(θ)=cosθ sec(θ)=secθ
Table 2

Recall that an odd function is one in which f(− x )= −f( x ) f(− x )= −f( x ) for all x x in the domain of f. f. The sine function is an odd function because sin( θ )=sinθ. sin( θ )=sinθ. The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs of π 2 π 2 and π 2 . π 2 . The output of sin( π 2 ) sin( π 2 ) is opposite the output of sin( π 2 ). sin( π 2 ). Thus,

sin( π 2 )=1 and sin( π 2 )=sin( π 2 ) =1 sin( π 2 )=1 and sin( π 2 )=sin( π 2 ) =1

This is shown in Figure 2.

Graph of y=sin(theta) from -2pi to 2pi, showing in particular that it is symmetric about the origin. Points given are (pi/2, 1) and (-pi/2, -1).
Figure 2 Graph of y=sinθ y=sinθ

Recall that an even function is one in which

f( x )=f( x ) for all x in the domain of f f( x )=f( x ) for all x in the domain of f

The graph of an even function is symmetric about the y-axis. The cosine function is an even function because cos(θ)=cosθ. cos(θ)=cosθ. For example, consider corresponding inputs π 4 π 4 and π 4 . π 4 . The output of cos( π 4 ) cos( π 4 ) is the same as the output of cos( π 4 ). cos( π 4 ). Thus,

cos( π 4 )=cos( π 4 )               0.707 cos( π 4 )=cos( π 4 )               0.707

See Figure 3.

Graph of y=cos(theta) from -2pi to 2pi, showing in particular that it is symmetric about the y-axis. Points given are (-pi/4, .707) and (pi/4, .707).
Figure 3 Graph of y=cosθ y=cosθ

For all θ θ in the domain of the sine and cosine functions, respectively, we can state the following:

  • Since sin(−θ )=sinθ, sin(−θ )=sinθ,sine is an odd function.
  • Since, cos(− θ )=cosθ, cos(− θ )=cosθ, cosine is an even function.

The other even-odd identities follow from the even and odd nature of the sine and cosine functions. For example, consider the tangent identity, tan(− θ )=−tanθ. tan(− θ )=−tanθ. We can interpret the tangent of a negative angle as tan(− θ )= sin( θ ) cos(− θ ) = sinθ cosθ =tanθ. tan(− θ )= sin( θ ) cos(− θ ) = sinθ cosθ =tanθ. Tangent is therefore an odd function, which means that tan( θ )=tan( θ ) tan( θ )=tan( θ ) for all θ θ in the domain of the tangent function.

The cotangent identity, cot( θ )=cotθ, cot( θ )=cotθ, also follows from the sine and cosine identities. We can interpret the cotangent of a negative angle as cot( θ )= cos( θ ) sin( θ ) = cosθ sinθ =cotθ. cot( θ )= cos( θ ) sin( θ ) = cosθ sinθ =cotθ. Cotangent is therefore an odd function, which means that cot( θ )=cot( θ ) cot( θ )=cot( θ ) for all θ θ in the domain of the cotangent function.

The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative angle will be interpreted as csc( θ )= 1 sin( θ ) = 1 sinθ =cscθ. csc( θ )= 1 sin( θ ) = 1 sinθ =cscθ. The cosecant function is therefore odd.

Finally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as sec( θ )= 1 cos( θ ) = 1 cosθ =secθ. sec( θ )= 1 cos( θ ) = 1 cosθ =secθ. The secant function is therefore even.

To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd, verifying the even-odd identities.

The next set of fundamental identities is the set of reciprocal identities, which, as their name implies, relate trigonometric functions that are reciprocals of each other. See Table 3.

Reciprocal Identities
sinθ= 1 cscθ sinθ= 1 cscθ cscθ= 1 sinθ cscθ= 1 sinθ
cosθ= 1 secθ cosθ= 1 secθ secθ= 1 cosθ secθ= 1 cosθ
tanθ= 1 cotθ tanθ= 1 cotθ cotθ= 1 tanθ cotθ= 1 tanθ
Table 3

The final set of identities is the set of quotient identities, which define relationships among certain trigonometric functions and can be very helpful in verifying other identities. See Table 4.

Quotient Identities
tanθ= sinθ cosθ tanθ= sinθ cosθ cotθ= cosθ sinθ cotθ= cosθ sinθ
Table 4

The reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions.

Summarizing Trigonometric Identities

The Pythagorean Identities are based on the properties of a right triangle.

cos 2 θ+ sin 2 θ=1 cos 2 θ+ sin 2 θ=1
1+ cot 2 θ= csc 2 θ 1+ cot 2 θ= csc 2 θ
1+ tan 2 θ= sec 2 θ 1+ tan 2 θ= sec 2 θ

The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle.

tan( θ )=tanθ tan( θ )=tanθ
cot( θ )=cotθ cot( θ )=cotθ
sin( θ )=sinθ sin( θ )=sinθ
csc( θ )=cscθ csc( θ )=cscθ
cos( θ )=cosθ cos( θ )=cosθ
sec( θ )=secθ sec( θ )=secθ

The reciprocal identities define reciprocals of the trigonometric functions.

sinθ= 1 cscθ sinθ= 1 cscθ
cosθ= 1 secθ cosθ= 1 secθ
tanθ= 1 cotθ tanθ= 1 cotθ
cscθ= 1 sinθ cscθ= 1 sinθ
secθ= 1 cosθ secθ= 1 cosθ
cotθ= 1 tanθ cotθ= 1 tanθ

The quotient identities define the relationship among the trigonometric functions.

tanθ= sinθ cosθ tanθ= sinθ cosθ
cotθ= cosθ sinθ cotθ= cosθ sinθ

Example 1

Graphing the Equations of an Identity

Graph both sides of the identity cotθ= 1 tanθ . cotθ= 1 tanθ . In other words, on the graphing calculator, graph y=cotθ y=cotθ and y= 1 tanθ . y= 1 tanθ .

Analysis

We see only one graph because both expressions generate the same image. One is on top of the other. This is a good way to confirm an identity verified with analytical means. If both expressions give the same graph, then they are most likely identities.

How To

Given a trigonometric identity, verify that it is true.

  1. Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.
  2. Look for opportunities to factor expressions, square a binomial, or add fractions.
  3. Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
  4. If these steps do not yield the desired result, try converting all terms to sines and cosines.

Example 2

Verifying a Trigonometric Identity

Verify tanθcosθ=sinθ. tanθcosθ=sinθ.

Analysis

This identity was fairly simple to verify, as it only required writing tanθ tanθ in terms of sinθ sinθ and cosθ. cosθ.

Try It #1

Verify the identity cscθcosθtanθ=1. cscθcosθtanθ=1.

Example 3

Verifying the Equivalency Using the Even-Odd Identities

Verify the following equivalency using the even-odd identities:

( 1+sinx )[ 1+sin( x ) ]= cos 2 x ( 1+sinx )[ 1+sin( x ) ]= cos 2 x

Example 4

Verifying a Trigonometric Identity Involving sec2θ

Verify the identity sec 2 θ1 sec 2 θ = sin 2 θ sec 2 θ1 sec 2 θ = sin 2 θ

Analysis

In the first method, we used the identity sec 2 θ= tan 2 θ+1 sec 2 θ= tan 2 θ+1 and continued to simplify. In the second method, we split the fraction, putting both terms in the numerator over the common denominator. This problem illustrates that there are multiple ways we can verify an identity. Employing some creativity can sometimes simplify a procedure. As long as the substitutions are correct, the answer will be the same.

Try It #2

Show that cotθ cscθ =cosθ. cotθ cscθ =cosθ.

Example 5

Creating and Verifying an Identity

Create an identity for the expression 2tanθsecθ 2tanθsecθ by rewriting strictly in terms of sine.

Example 6

Verifying an Identity Using Algebra and Even/Odd Identities

Verify the identity:

sin 2 ( θ ) cos 2 ( θ ) sin( θ )cos( θ ) =cosθsinθ sin 2 ( θ ) cos 2 ( θ ) sin( θ )cos( θ ) =cosθsinθ
Try It #3

Verify the identity sin 2 θ1 tanθsinθtanθ = sinθ+1 tanθ . sin 2 θ1 tanθsinθtanθ = sinθ+1 tanθ .

Example 7

Verifying an Identity Involving Cosines and Cotangents

Verify the identity: ( 1 cos 2 x )( 1+ cot 2 x )=1. ( 1 cos 2 x )( 1+ cot 2 x )=1.

Using Algebra to Simplify Trigonometric Expressions

We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.

For example, the equation ( sinx+1 )( sinx1 )=0 ( sinx+1 )( sinx1 )=0 resembles the equation ( x+1 )( x1 )=0, ( x+1 )( x1 )=0, which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations.

Another example is the difference of squares formula, a 2 b 2 =( ab )( a+b ), a 2 b 2 =( ab )( a+b ), which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve.

Example 8

Writing the Trigonometric Expression as an Algebraic Expression

Write the following trigonometric expression as an algebraic expression: 2 cos 2 θ+cosθ1. 2 cos 2 θ+cosθ1.

Example 9

Rewriting a Trigonometric Expression Using the Difference of Squares

Rewrite the trigonometric expression: 4 cos 2 θ1. 4 cos 2 θ1.

Analysis

If this expression were written in the form of an equation set equal to zero, we could solve each factor using the zero factor property. We could also use substitution like we did in the previous problem and let cosθ=x, cosθ=x, rewrite the expression as 4 x 2 1, 4 x 2 1, and factor ( 2x1 )( 2x+1 ). ( 2x1 )( 2x+1 ). Then replace x x with cosθ cosθ and solve for the angle.

Try It #4

Rewrite the trigonometric expression: 259 sin 2 θ. 259 sin 2 θ.

Example 10

Simplify by Rewriting and Using Substitution

Simplify the expression by rewriting and using identities:

csc 2 θ cot 2 θ csc 2 θ cot 2 θ
Try It #5

Use algebraic techniques to verify the identity: cosθ 1+sinθ = 1sinθ cosθ . cosθ 1+sinθ = 1sinθ cosθ .

(Hint: Multiply the numerator and denominator on the left side by 1sinθ.) 1sinθ.)

Media

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

7.1 Section Exercises

Verbal

1.

We know g(x)=cosx g(x)=cosx is an even function, and f(x)=sinx f(x)=sinx and h(x)=tanx h(x)=tanx are odd functions. What about G(x)= cos 2 x,F(x)= sin 2 x, G(x)= cos 2 x,F(x)= sin 2 x, and H(x)= tan 2 x? H(x)= tan 2 x? Are they even, odd, or neither? Why?

2.

Examine the graph of f(x)=secx f(x)=secx on the interval [π,π]. [π,π]. How can we tell whether the function is even or odd by only observing the graph of f(x)=secx? f(x)=secx?

3.

After examining the reciprocal identity for sect, sect, explain why the function is undefined at certain points.

4.

All of the Pythagorean Identities are related. Describe how to manipulate the equations to get from sin 2 t+ cos 2 t=1 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.

5.

sinxcosxsecx sinxcosxsecx

6.

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

7.

tanxsinx+secx cos 2 x tanxsinx+secx cos 2 x

8.

cscx+cosxcot(x) cscx+cosxcot(x)

9.

cott+tant sec(t) cott+tant sec(t)

10.

3 sin 3 tcsct+ cos 2 t+2cos(t)cost 3 sin 3 tcsct+ cos 2 t+2cos(t)cost

11.

tan(x)cot(x) tan(x)cot(x)

12.

sin(x)cosxsecxcscxtanx cotx sin(x)cosxsecxcscxtanx cotx

13.

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

14.

( tanx csc 2 x + tanx sec 2 x )( 1+tanx 1+cotx ) 1 cos 2 x ( tanx csc 2 x + tanx sec 2 x )( 1+tanx 1+cotx ) 1 cos 2 x

15.

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

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

16.

tanx+cotx cscx ;cosx tanx+cotx cscx ;cosx

17.

secx+cscx 1+tanx ;sinx secx+cscx 1+tanx ;sinx

18.

cosx 1+sinx +tanx;cosx cosx 1+sinx +tanx;cosx

19.

1 sinxcosx cotx;cotx 1 sinxcosx cotx;cotx

20.

1 1cosx cosx 1+cosx ;cscx 1 1cosx cosx 1+cosx ;cscx

21.

( secx+cscx )( sinx+cosx )2cotx;tanx ( secx+cscx )( sinx+cosx )2cotx;tanx

22.

1 cscxsinx ;secx and tanx 1 cscxsinx ;secx and tanx

23.

1sinx 1+sinx 1+sinx 1sinx ;secx and tanx 1sinx 1+sinx 1+sinx 1sinx ;secx and tanx

24.

tanx;secx tanx;secx

25.

secx;cotx secx;cotx

26.

secx;sinx secx;sinx

27.

cotx;sinx cotx;sinx

28.

cotx;cscx cotx;cscx

For the following exercises, verify the identity.

29.

cosx cos 3 x=cosx sin 2 x cosx cos 3 x=cosx sin 2 x

30.

cosx( tanxsec( x ) )=sinx1 cosx( tanxsec( x ) )=sinx1

31.

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

32.

( sinx+cosx ) 2 =1+2sinxcosx ( sinx+cosx ) 2 =1+2sinxcosx

33.

cos 2 x tan 2 x=2 sin 2 x sec 2 x cos 2 x tan 2 x=2 sin 2 x sec 2 x

Extensions

For the following exercises, prove or disprove the identity.

34.

1 1+cosx 1 1cos(x) =2cotxcscx 1 1+cosx 1 1cos(x) =2cotxcscx

35.

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

36.

( sec 2 (x) tan 2 x tanx )( 2+2tanx 2+2cotx )2 sin 2 x=cos2x ( sec 2 (x) tan 2 x tanx )( 2+2tanx 2+2cotx )2 sin 2 x=cos2x

37.

tanx secx sin( x )= cos 2 x tanx secx sin( x )= cos 2 x

38.

sec( x ) tanx+cotx =sin( x ) sec( x ) tanx+cotx =sin( x )

39.

1+sinx cosx = cosx 1+sin( x ) 1+sinx cosx = cosx 1+sin( x )

For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.

40.

cos 2 θ sin 2 θ 1 tan 2 θ = sin 2 θ cos 2 θ sin 2 θ 1 tan 2 θ = sin 2 θ

41.

3 sin 2 θ+4 cos 2 θ=3+ cos 2 θ 3 sin 2 θ+4 cos 2 θ=3+ cos 2 θ

42.

secθ+tanθ cotθ+cosθ = sec 2 θ secθ+tanθ cotθ+cosθ = sec 2 θ

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