Sec. 2.7 Proving Trigonometric Identities 87 Section 2.7 Proving Trigonometric Identities In this section, we use the identities presented in Section 2.6 to do two different tasks: ) to simplify a trigonometric expression, and 2) prove that a supposed identity really is an identity. SIMPLIFYING A TRIGONOMETRIC EXPRESSION Example : Write each expression in terms of and/or, and simplify. a) secθ cotθ b) tanθ + secθ Use the reciprocal and ratio identities to first write the expression in terms of only and, and then simplify, if possible. Answer: a) secθ cotθ Write each in terms of sine and cosine. Divide out. This is the same as cscθ. cscθ This expression cannot be simplified further. b) tanθ + secθ Write each in terms of sine and cosine. + + Multiply to the second fraction. These fractions have the same denominator; we can add them directly. 2 can simplify to tanθ. 2tanθ This expression cannot be simplified further. Sec. 2.7 Proving Trigonometric Identities 87 Robert H. Prior, 206
88 Sec. 2.7 Proving Trigonometric Identities You Try It Write each expression in terms of and/or, and simplify. a) secθ tanθ b) cscθ cotθ In Example b) we added two fractions with the same denominator. If the fractions have different denominators, then we must first identify the least common denominator (LCD) and build up each fraction appropriately. Example 2: Build up each fraction to have common denominators and simplify. Identify the LCD and multiply by a form of to create the LCD in each fraction. Answer:. The LCD is. Multiply each by a form of to build up each fraction. Simplify the respective fractions. Combine the fractions into one fraction. This expression cannot be simplified further. Sec. 2.7 Proving Trigonometric Identities 88 Robert H. Prior, 206
Sec. 2.7 Proving Trigonometric Identities 89 Example 3: In each expression build up the fractions to have a common denominator. Simplify. a) b) sin 2 θ + cos 2 θ. Identify the LCD and build up each fraction to have that denominator. Answer: a) The LCD is. Write a in the denominator of the second fraction and multiply by a form of to build it up. cos2 θ Multiply. Combine the fractions into one. cos 2 θ The numerator is sin2 θ. sin 2 θ This expression is one simplified form. Other simplified forms are possible. b) sin 2 θ + cos 2 θ. The LCD is sin 2 θ cos 2 θ. Multiply each by a form of to build up the fractions. sin 2 θ cos2 θ cos 2 θ + cos 2 θ. sin2 θ sin 2 θ cos 2 θ sin 2 θ cos 2 θ + cos 2 θ + sin 2 θ sin 2 θ cos 2 θ sin 2 θ cos 2 θ sin 2 θ sin 2 θ cos 2 θ Simplify the respective fractions. Combine the fractions into one fraction. The numerator is a Pythagorean identity; replace the numerator with. Sec. 2.7 Proving Trigonometric Identities 89 Robert H. Prior, 206
90 Sec. 2.7 Proving Trigonometric Identities You Try It 2 In each expression build up the fractions to have a common denominator. Simplify. Hint: In expression b), the second fraction has a denominator of. a) + b) cos 2 θ Example 4: Multiply and simplify. a) ( 2) b) (2 3)( + 4) For each, distribute. You may use FOIL for b). Answer: a) ( 2) b) (2 3)( + 4) 2 2cos 2 θ + 8 3 2 2cos 2 θ + 5 2 You Try It 3 Multiply and simplify. a) (cscθ ) b) ( + ) 2 Sec. 2.7 Proving Trigonometric Identities 90 Robert H. Prior, 206
Sec. 2.7 Proving Trigonometric Identities 9 PROVING IDENTITIES An identity is set up to look like an equation. However, when we are attempting to prove an identity (a supposed identity) is true, then we cannot assume that the left and right sides are equivalent, so we are not allowed to treat them like equations. In other words, we cannot use rules that apply to equations, such as adding the same term to both sides of the equal sign. Instead, we must manipulate only one side of the supposed identity to make it identical to the other side. Once that task is complete, the identity has been proven. Note: Once an identity is proven, then we can write alternative identities using the rules of equations. For example, because we know that cos 2 θ + sin 2 θ... we can solve for sin 2 θ: sin 2 θ cos 2 θ Example 5: For each, demonstrate that the equation is an identity by transforming the left side (only) to be equivalent to the right side. a) cscθ tanθ secθ b) tanθ secθ Start by writing the left side in terms of sine and cosine only. Answer: a) cscθ tanθ secθ Write the left side in terms of sine and cosine. Divide out. This is secθ. Finish the proof. secθ secθ QED* b) tanθ secθ Write the left side in terms of sine and cosine. Change division to multiplying by the reciprocal. Divide out, and finish the proof. QED* *Note: It is common though not required to write QED at the end of a proof. It stands for the Latin phrase Quod Erat Demonstrandum; loosely translated, it says, that which was to be demonstrated. Sec. 2.7 Proving Trigonometric Identities 9 Robert H. Prior, 206
92 Sec. 2.7 Proving Trigonometric Identities Some proofs are more involved and require using a variety of identities, including various forms of the Pythagorean identity. Example 6: Demonstrate that the equation is an identity by transforming the left side (only) to be equivalent to the right side. cscθ tanθ sin2 θ Start by writing the left side in terms of sine and cosine only. Answer: a) cscθ tanθ sin2 θ Write the left side in terms of sine and cosine. Simplify the first fractions and write the second term as a fraction with in the denominator. Get the common denominator: LCD. cos2 θ Simplify the second fraction. We must combine the fractions before we can use any other identities. cos 2 θ We can now use a Pythagorean identity in the numerator and finish the proof. sin 2 θ sin2 θ QED You Try It Answers YTI : a) tan 2 θ b) YTI 2: a) b) cos 2 θ sin 2 θ (This cannot simplify further at this time.) YTI 3: a) cos 2 θ b) cos 2 θ + 2 + Sec. 2.7 Proving Trigonometric Identities 92 Robert H. Prior, 206
Sec. 2.7 Proving Trigonometric Identities 93 Section 2.7 Focus Exercises First write each expression in terms of only and/or. Then simplify.. tanθ 2. tanθ 3. cscθ 4. cscθ tanθ 5. cotθ secθ 6. tanθ 7. tanθ + secθ 8. cotθ + cscθ 9. secθ tanθ 0. secθ tanθ Sec. 2.7 Proving Trigonometric Identities 93 Robert H. Prior, 206
94 Sec. 2.7 Proving Trigonometric Identities For each, identify the least common denominator (LCD) and use it to build up each fraction and combine the fractions. Write all answers in terms of and. Simplify.. + cos 2 θ. 2. 3.. 4. +. Multiply and simplify 5. (tanθ ) 6. cscθ (tanθ + ) Sec. 2.7 Proving Trigonometric Identities 94 Robert H. Prior, 206
Sec. 2.7 Proving Trigonometric Identities 95 7. ( + 2)( + ) 8. ( + ) 2 For each, demonstrate that the equation is an identity by transforming the left side (only) to be equivalent to the right side. 9. secθ 20. secθ cotθ cscθ 2. cotθ 22. secθ cscθ tanθ Sec. 2.7 Proving Trigonometric Identities 95 Robert H. Prior, 206
96 Sec. 2.7 Proving Trigonometric Identities 23. sin 2 θ tanθ 24. cscθ cotθ 25. cscθ + secθ 26. tanθ secθ Sec. 2.7 Proving Trigonometric Identities 96 Robert H. Prior, 206