5.2 Verifying Trigonometric Identities

333202 0502.qxd38212/5/05Chapter 55.29:01 AMPage 382Analytic TrigonometryVerifying Trigonometric IdentitiesWhat you should learn Verify trigonometric identities.Why you should learn itYou can use trigonometric identities to rewrite trigonometricequations that model real-lifesituations. For instance, inExercise 56 on page 388, youcan use trigonometric identitiesto simplify the equation thatmodels the length of a shadowcast by a gnomon (a device usedto tell time).IntroductionIn this section, you will study techniques for verifying trigonometric identities.In the next section, you will study techniques for solving trigonometric equations. The key to verifying identities and solving equations is the ability to usethe fundamental identities and the rules of algebra to rewrite trigonometricexpressions.Remember that a conditional equation is an equation that is true for onlysome of the values in its domain. For example, the conditional equationsin x 0Conditional equationis true only for x n , where n is an integer. When you find these values, youare solving the equation.On the other hand, an equation that is true for all real values in the domainof the variable is an identity. For example, the familiar equationsin2 x 1 cos 2 xIdentityis true for all real numbers x. So, it is an identity.Verifying Trigonometric IdentitiesAlthough there are similarities, verifying that a trigonometric equation is anidentity is quite different from solving an equation. There is no well-defined setof rules to follow in verifying trigonometric identities, and the process is bestlearned by practice.Guidelines for Verifying Trigonometric Identities1. Work with one side of the equation at a time. It is often better to workwith the more complicated side first.Robert Ginn/PhotoEdit2. Look for opportunities to factor an expression, add fractions, square abinomial, or create a monomial denominator.3. Look for opportunities to use the fundamental identities. Note whichfunctions are in the final expression you want. Sines and cosines pair upwell, as do secants and tangents, and cosecants and cotangents.4. If the preceding guidelines do not help, try converting all terms to sinesand cosines.5. Always try something. Even paths that lead to dead ends provide insights.You may want to review the distinctionsamong expressions, equations, andidentities. Have your students look atsome algebraic identities and conditionalequations before starting this section.It is important for them to understandwhat it means to verify an identity andnot try to solve it as an equation.Verifying trigonometric identities is a useful process if you need to converta trigonometric expression into a form that is more useful algebraically. Whenyou verify an identity, you cannot assume that the two sides of the equation areequal because you are trying to verify that they are equal. As a result, whenverifying identities, you cannot use operations such as adding the same quantityto each side of the equation or cross multiplication.

333202 0502.qxd12/5/059:01 AMPage 383Section 5.2Example 1Verifying Trigonometric Identities383Verifying a Trigonometric IdentityVerify the identitysec2 1 sin2 .sec2 SolutionBecause the left side is more complicated, start with it.Remember that an identity isonly true for all real values inthe domain of the variable.For instance, in Example 1the identity is not true when 2 because sec2 isnot defined when 2.sec2 1 tan2 1 1 sec2 sec2 tan2 sec2 Simplify. tan2 cos 2 Pythagorean identitysin2 cos2 cos2 sin2 Reciprocal identityQuotient identitySimplify.Notice how the identity is verified. You start with the left side of the equation (themore complicated side) and use the fundamental trigonometric identities tosimplify it until you obtain the right side.Now try Exercise 5.Encourage your students to identifythe reasoning behind each solutionstep in the examples of this sectionwhile covering the comment lines.This will help students to recognizeand remember the fundamentaltrigonometric identities.There is more than one way to verify an identity. Here is another way toverify the identity in Example 1.sec2 1 sec2 1 22sec sec sec2 Example 2Rewrite as the difference of fractions. 1 cos 2 Reciprocal identity sin2 Pythagorean identityCombining Fractions Before Using IdentitiesVerify the identity11 2 sec2 .1 sin 1 sin Solution111 sin 1 sin 1 sin 1 sin 1 sin 1 sin Add fractions. 21 sin2 Simplify. 2cos2 Pythagorean identity 2 sec2 Now try Exercise 19.Reciprocal identity

333202 0502.qxd38412/5/05Chapter 5Example 39:01 AMPage 384Analytic TrigonometryVerifying Trigonometric IdentityVerify the identity tan2 x 1 cos 2 x 1 tan2 x.Algebraic SolutionNumerical SolutionBy applying identities before multiplying, you obtain the following.Use the table feature of a graphing utility set inradian mode to create a table that shows thevalues of y1 tan2 x 1 cos2 x 1 andy2 tan2 x for different values of x, as shownin Figure 5.2. From the table you can see that thevalues of y1 and y2 appear to be identical, so tan2 x 1 cos2 x 1 tan2 x appears tobe an identity. tan2 x 1 cos 2 x 1 sec2 x sin2 x sin2 xcos 2 x cos x sin xPythagorean identitiesReciprocal identity2 tan2 xRule of exponentsQuotient identityNow try Exercise 39.FIGUREExample 45.2Converting to Sines and CosinesVerify the identity tan x cot x sec x csc x.SolutionTry converting the left side into sines and cosines.Although a graphing utility canbe useful in helping to verify anidentity, you must use algebraictechniques to produce a validproof.cos xsin x cos xsin xQuotient identities sin2 x cos 2 xcos x sin xAdd fractions. 1cos x sin xPythagorean identity 1cos xReciprocal identitiestan x cot x 1 sin x sec x csc xNow try Exercise 29.As shown at the right,csc2 x 1 cos x is considered asimplified form of 1 1 cos x because the expression does notcontain any fractions.Recall from algebra that rationalizing the denominator using conjugates is, onoccasion, a powerful simplification technique. A related form of this technique,shown below, works for simplifying trigonometric expressions as well.111 cos x1 cos x1 cos x 1 cos x 1 cos x 1 cos x1 cos2 xsin2 x csc2 x 1 cos x This technique is demonstrated in the next example.

333202 0502.qxd12/5/059:01 AMPage 385Section 5.2Verifying Trigonometric Identities385Verifying Trigonometric IdentitiesExample 5Verify the identity sec y tan y cos y.1 sin ySolutionBegin with the right side, because you can create a monomial denominator bymultiplying the numerator and denominator by 1 sin y.cos ycos y1 sin y 1 sin y 1 sin y 1 sin y cos y cos y sin y1 sin2 ycos y cos y sin y cos 2 ycos ycos y sin y 2cos ycos2 y1sin y cos y cos y sec y tan yMultiply numerator anddenominator by 1 sin y.Multiply.Pythagorean identityWrite as separate fractions.Simplify.IdentitiesNow try Exercise 33.In Examples 1 through 5, you have been verifying trigonometric identities byworking with one side of the equation and converting to the form given on theother side. On occasion, it is practical to work with each side separately, to obtainone common form equivalent to both sides. This is illustrated in Example 6.Example 6Working with Each Side SeparatelyVerify the identitycot 2 1 sin .1 csc sin SolutionWorking with the left side, you havecot 2 csc2 1 1 csc 1 csc csc 1 csc 1 1 csc csc 1.Pythagorean identityFactor.Simplify.Now, simplifying the right side, you have1 sin 1sin sin sin sin csc 1.Write as separate fractions.Reciprocal identityThe identity is verified because both sides are equal to csc 1.Now try Exercise 47.

333202 0502.qxd38612/5/05Chapter 59:01 AMPage 386Analytic TrigonometryIn Example 7, powers of trigonometric functions are rewritten as morecomplicated sums of products of trigonometric functions. This is a commonprocedure used in calculus.Example 7Three Examples from CalculusVerify each identity.a. tan4 x tan2 x sec2 x tan2 xb. sin3 x cos4 x cos4 x cos 6 x sin xc. csc4 x cot x csc2 x cot x cot3 x Solutiona. tan4 x tan2 x tan2 x tan2 x sec2 x 1 Write as separate factors.Pythagorean identity tan2 x sec2 x tan2 xMultiply.b. sin3 x cos4 x sin2 x cos4 x sin x 1 cos2 x cos4 x sin x cos4x cos6x sin xc. csc4 x cot x csc2 x csc2 x cot x csc2 x 1 cot2 x cot x csc2 x cot x cot3 x Write as separate factors.Pythagorean identityMultiply.Write as separate factors.Pythagorean identityMultiply.Now try Exercise 49.WAlternative Writing AboutMathematicsa. Ask students to assemble a list oftechniques and strategies forrewriting trigonometric expressionssuch as those demonstrated in theexamples of this section.b. Ask students to work in pairs. Eachstudent should create an identityequation from the fundamentaltrigonometric identities. Partnersthen trade and verify one another’sidentities. Then have students write abrief explanation of the techniquesthey used to create the identities.RITING ABOUTMATHEMATICSError Analysis You are tutoring a student in trigonometry. One of the homeworkproblems your student encounters asks whether the following statement is anidentity.? 5tan2 x sin2 x tan2 x6Your student does not attempt to verify the equivalence algebraically, but mistakenly uses only a graphical approach. Using range settings ofXmin 3 Ymin 20Xmax 3 Ymax 20Xscl 2Yscl 1your student graphs both sides of the expression on a graphing utility andconcludes that the statement is an identity.What is wrong with your student’s reasoning? Explain. Discuss the limitationsof verifying identities graphically.

333202 0502.qxd12/5/059:01 AMPage 387Section 5.25.2Verifying Trigonometric Identities387ExercisesVOCABULARY CHECK:In Exercises 1 and 2, fill in the blanks.1. An equation that is true for all real values in its domain is called an .2. An equation that is true for only some values in its domain is called a .In Exercises 3–8, fill in the blank to complete the trigonometric identity.3.1 cot u4.cos u sin u 2 u 5. sin2 u 16. cos7. csc u 8. sec u PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.In Exercises 1–38, verify the identity.1. sin t csc t 123.2. sec y cos y 13. 1 sin 1 sin cos 2 11 1sin x 1 csc x 124. cos x 4. cot 2 y sec 2 y 1 15. cos 2 sin2 1 2 sin2 25. tan6. cos 2 sin2 2 cos 2 17. sin2 sin4 cos 2 cos4 8. cos x sin x tan x sec xcsc2 csc sec 9.cot 11.cot2 t csc t sin tcsc tcot3 t cos t csc2 t 1 10.csc t12.1sec2 tan tan tan 27.cos 2 x tan xsin 2 xcsc x cot xsec x tan x cot x sec xcos x30.tan x tan ycot x cot y 1 tan x tan y cot x cot y 131.tan x cot y tan y cot xtan x cot ycos x cos ysin x sin y 0sin x sin ycos x cos y1 csc x sin xsec x tan x32.16.sec 1 sec 1 cos 33.18. sec x cos x sin x tan x26.29.15.17. csc x sin x cos x cot x 2 tan 128. 1 sin y 1 sin y cos2 y13. sin1 2 x cos x sin5 2 x cos x cos3 x sin x14. sec6 x sec x tan x sec4 x sec x tan x sec5 x tan3 xcos xsin x cos x 1 tan x sin x cos x 1 sin 11 sinsin cos 1 cos 1 cos 34. 1 cos sin 19.11 tan x cot xtan x cot x20.11 csc x sin xsin x csc x36. sec2 y cot 221.cos cot 1 csc 1 sin 37. sin t csc22.cos 1 sin 2 sec cos 1 sin 38. sec235. cos2 cos2 2 1 2 y 1 2 t tan t 2 x 1 cot2x