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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?

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