Lesson 5

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.