Lesson 5

Lesson 5: Proving Trigonometric Identities

Focus

When a person decides to buy a vehicle, or a mechanic needs to fix a vehicle, that person will often reference the technical specifications for that vehicle. Specifications give details of things like vehicle size, parts, cost, and performance. But before the specifications are written, the car must go through a design process. Engineers spend a lot of time and thought trying to make the car appealing and functional.

An identity is like the specifications of a vehicle. An identity is a ready tool that a user can expect to give reliable information without a lot of thought. Just like a car must be designed before the specifics can be used, an identity must be proven before the identiry is used; the identity must go through a testing procedure to show it will always work properly. This testing procedure is called a proof.

Lesson Outcome

At the end of this lesson you will be able to prove a trigonometric identity.

Lesson Questions

In this lesson you will investigate the following questions:

• How can a trigonometric identity be proven?
• Why is it necessary to prove an identity?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 5 Assignment (Download the Lesson 5 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms
• work under Project Connection

Discover

Try This 1

Consider the possible identity = x + 150.

1. Verify = x + 150 for x = −81, 0, and 176.
2. Verify = x + 150 graphically for −10 < x < 10.
3. Is = x + 150 an identity?
4. Try using x = −200 to verify = x + 150. Does your answer to question 3 change?

Save your answers in your course folder.

Share 1

With a partner or group, discuss the following questions based on Try This 1.

1. What limitations are there to verifying an identity numerically or graphically?
2. It would be impossible to verify an identity like this for all possible inputs. How could you determine whether an identity is true for all defined values?

If required, save a record of your discussion in your course folder.

Explore

In Try This 1 you saw there are limitations to verifying an identity numerically and graphically. It is impossible to test every possible value for the variable in most potential identities. This leads to the idea of a mathematical proof. A proof is an argument that shows how you know a statement is true. The statement is called a theorem once the statement has been proven. The theorem can then be used to solve problems or prove other statements. Complete Try This 2 to explore the proof of a trigonometric identity.

Try This 2

Consider the potential identity .

1. Determine the non-permissible values for the potential identity.
2.  
   1. Verify the identity for and −1.5.
   2. Verify the identity graphically.
3. Show that  simplifies to .
4. Show that sec x simplifies to .
   

The answers to questions 3 and 4 together are a proof of the identity = sec x for all permissible values. This means you know  is true for all permissible values of x.

5. Try to solve = sec x for x. What do you notice?
   

Save your answers in your course folder.

Share 2

With a partner or group, discuss the following questions based on the information from Try This 2.

1. When is the identity valid?
2. Why do the answers to questions 3 and 4 of Try This 2 prove that = sec x is an identity?
3. Why doesn’t the answer to question 5 of Try This 2 prove that = sec x is an identity?

In Try This 2 you showed that it is possible to express each side of  in the equivalent form Reducing each side to a common expression is a common style of proof used for trigonometric identities. In question 5 of Try This 2, you tried to solve the equation  and probably reduced it to something like 1 = 1 or 0 = 0.

Although solving an identity gives a true statement, this is not a proof. Consider the equation  from the beginning of the lesson. It is also possible to simplify this to something true, but you know it is not an identity because reduces to |x + 150|, not x + 150.

Proving an identity does not follow a set pattern; each proof is unique. Here are some strategies that may be useful.

• Only manipulate one side at a time.
• Begin with the more complex side; it is often easier to simplify than to expand.
• Use known identities to make substitutions.
• Try changing all trigonometric ratios to sines and cosines.
• When second-degree trigonometric functions are used (sin2 x, tan2 x), consider Pythagorean identities.
• When the inputs for the functions are different (cos 2x, cos x), consider double-angle, sum, or difference identities.
• Try factoring an expression.
• Try expanding an expression.
• If fractions are used, try writing each side as a single fraction.
• Try multiplying the numerator and denominator by the conjugate of an expression. This can be helpful if 1 ± sin x or 1 ± cos x appears as part of a fraction. (The conjugate of A + B is A − B.)

The most important suggestion of all is DON’T GIVE UP. You will rarely write a proof correctly on your first try. Proving an identity will often require multiple attempts, a lot of scrap paper, and a bit of luck—none of which are shown in examples of proofs. Be patient.

Complete Try This 3 with a group if possible, but alone if necessary.

Try This 3

Consider the identity csc x(1 + sin x) = 1 + csc x.

1. What values of x are not permissible for this identity?
2. Which strategies do you think may be used to prove the identity?
3. Try using one or more of the strategies you discussed in question 2 to prove the identity.

4. How many different proofs are possible? Explain.

5. How do you know when you are finished a proof?

Save your responses in your course folder.

Share 3

Compare your proof to the proofs of other students. Your comparisons can include considerations like length, difficulty, clarity, and the strategy used.

If required, save your responses and a record of your discussion in your course folder.

There are many different ways of writing the proof of an identity correctly; however, the proof style used in this lesson always ends with the left side the same as the right side. Two possible proofs are shown in Example Proof 1A and Example Proof 1B. Compare these proofs to your proof.

Self-Check 2

Self-Check 3

Compare your proof from Try This 5 to Example Proof 3. How do the strategies compare?

Self-Check 4

Connect

Lesson 5 Assignment

Lesson 5 Summary

In this lesson you learned about mathematical proofs, which are at the core of mathematics. A proof is just an argument showing that a statement must be true. Once you know the statement is always true, you can use the statement in future scenarios, such as simplifying expressions and solving other equations.

For the type of proof you learned in this lesson, the goal is to show that both sides of the identity are equal by manipulating each side independently until they are the same expression. There is no specific pattern for how to proceed with a given proof. A bit of luck, determination, and some intuition will help you complete a proof.

In the next module you will begin to explore exponents and their inverse, logarithms.