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OpenStax-CNX module: m49393 1 Solving Trigonometric Equations with Identities ∗ OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract In this section, you will: • • ∗ † Verify the fundamental trigonometric identities. Simplify trigonometric expressions using algebra and the identities. Version 1.5: Mar 11, 2015 10:47 am +0000 http://creativecommons.org/licenses/by/4.0/ http://https://legacy.cnx.org/content/m49393/1.5/ † OpenStax-CNX module: m49393 Figure 1: 2 International passports and travel documents In espionage movies, we see international spies with multiple passports, each claiming a dierent identity. However, we know that each of those passports represents the same person. The trigonometric identities act in a similar manner to multiple passportsthere are many ways to represent the same trigonometric expression. Just as a spy will choose an Italian passport when traveling to Italy, we choose the identity that applies to the given scenario when solving a trigonometric equation. In this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions. 1 Verifying the Fundamental Trigonometric Identities Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, nding common denominators, and using special formulas are the basic tools of solving algebraic equations. In fact, we use algebraic techniques constantly to simplify trigonometric expressions. Basic properties and formulas of algebra, such as the dierence of squares formula and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations. We already know that all of the trigonometric functions are related because they all are dened in terms of the unit circle. Consequently, any trigonometric identity can be written in many ways. http://https://legacy.cnx.org/content/m49393/1.5/ OpenStax-CNX module: m49393 3 To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation. Sometimes we have to factor expressions, expand expressions, nd common denominators, or use other algebraic strategies to obtain the desired result. In this rst section, we will work with the fundamental identities: the Pythagorean identities, the even-odd identities, the reciprocal identities, and the quotient identities. We will begin with the Pythagorean identities (see Table 1), which are equations involving trigonometric functions based on the properties of a right triangle. We have already seen and used the rst of these identies, but now we will also use additional identities. Pythagorean Identities 2 2 sin θ + cos θ = 1 1 + cot2 θ = csc2 θ 1 + tan2 θ = sec2 θ Table 1 The second and third identities can be obtained by manipulating the rst. The identity 1+cot θ = csc θ is found by rewriting the left side of the equation in terms of sine and cosine. Prove: 1 + cot θ = csc θ 1 + cot θ = 1 + Rewrite the left side. + Write both terms with the common denominator. = (1) = 2 2 2 2 cos2 θ sin2 θ 2 sin2 θ sin2 θ = cos2 θ sin2 θ sin2 θ+cos2 θ sin2 θ 1 sin2 θ 2 = csc θ Similarly, 1 + tan θ = sec θ can be obtained by rewriting the left side of this identity in terms of sine and cosine. This gives 1 + tan θ = 1 + Rewrite left side. + Write both terms with the common denominator. = (2) = 2 2 2 = sin θ 2 cos θ sin θ 2 cos θ 2 cos θ cos θ cos2 θ+sin2 θ cos2 θ 1 cos2 θ 2 = sec θ The next set of fundamental identities is the set of even-odd identities. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity is odd or even. (See Table 2). Even-Odd Identities tan (−θ) = −tan θ sin (−θ) = −sin θ cos (−θ) = cos θ cot (−θ) = −cot θ csc (−θ) = −csc θ sec (−θ) = sec θ Table 2 http://https://legacy.cnx.org/content/m49393/1.5/ OpenStax-CNX module: m49393 4 Recall that an odd function is one in which f (−x) = −f (x) for all x in the domain of f. The sine function is an odd function because sin (−θ) = −sin θ. The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs of and − . The output of sin is opposite the output of sin − . Thus, π 2 π 2 π 2 =1 − π2 = −sin sin sin Graph of π 2 and π 2 (3) = −1 This is shown in Figure 2. Figure 2: π 2 y = sin θ Recall that an even function is one in which f (−x) = f (x) for all x in the domain of f (4) The graph of an even function is symmetric about the axis. The cosine function is an even function because cos (−θ) = cos θ. For example, consider corresponding inputs and − . The output of cos is the same as the output of cos − . Thus, y- π 4 π 4 cos − π4 = cos See Figure 3. http://https://legacy.cnx.org/content/m49393/1.5/ π 4 ≈ 0.707 π 4 π 4 (5) OpenStax-CNX module: m49393 Figure 3: Graph of 5 y = cos θ For all θ in the domain of the sine and cosine functions, respectively, we can state the following: • Since sin (−θ) = −sin θ, sine is an odd function. • Since, cos (−θ) = cos θ, cosine is an even function. The other even-odd identities follow from the even and odd nature of the sine and cosine functions. For example, consider the tangent identity, tan (−θ) = −tan θ. We can interpret the tangent of a negative angle as tan (−θ) = = = −tan θ. Tangent is therefore an odd function, which means that tan (−θ) = −tan (θ) for all θ in the domain of the tangent function. The cotangent identity, cot (−θ) = −cot θ, also follows from the sine and cosine identities. We can interpret the cotangent of a negative angle as cot (−θ) = = = −cot θ. Cotangent is therefore an odd function, which means that cot (−θ) = −cot (θ) for all θ in the domain of the cotangent function. The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative = = −csc θ. The cosecant function is therefore odd. angle will be interpreted as csc (−θ) = Finally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as sec (−θ) = = = sec θ. The secant function is therefore even. To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd, verifying the even-odd identities. The next set of fundamental identities is the set of reciprocal identities, which, as their name implies, relate trigonometric functions that are reciprocals of each other. See Table 3. sin(−θ) cos(−θ) −sin θ cos θ cos(−θ) sin(−θ) 1 sin(−θ) 1 cos(−θ) cos θ −sin θ 1 −sin θ 1 cos θ Reciprocal Identities sin θ = cos θ = tan θ = http://https://legacy.cnx.org/content/m49393/1.5/ 1 csc θ 1 sec θ 1 cot θ csc θ = sec θ = cot θ = 1 sin θ 1 cos θ 1 tan θ OpenStax-CNX module: m49393 6 Table 3 The nal set of identities is the set of quotient identities, which dene relationships among certain trigonometric functions and can be very helpful in verifying other identities. See Table 4. Quotient Identities tan θ = sin θ cos θ cot θ = cos θ sin θ Table 4 The reciprocal and quotient identities are derived from the denitions of the basic trigonometric functions. The Pythagorean identities are based on the properties of a right triangle. cos θ + sin θ = 1 (6) 1 + cot θ = csc θ (7) 1 + tan θ = sec θ (8) The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle. tan (−θ) = −tan θ (9) cot (−θ) = −cot θ (10) sin (−θ) = −sin θ (11) csc (−θ) = −csc θ (12) cos (−θ) = cos θ (13) sec (−θ) = sec θ (14) The reciprocal identities dene reciprocals of the trigonometric functions. A General Note: 2 2 http://https://legacy.cnx.org/content/m49393/1.5/ 2 2 2 2 sin θ = 1 csc θ (15) cos θ = 1 sec θ (16) tan θ = 1 cot θ (17) csc θ = 1 sin θ (18) OpenStax-CNX module: m49393 7 sec θ = 1 cos θ (19) cot θ = 1 tan θ (20) tan θ = sin θ cos θ (21) cot θ = cos θ sin θ (22) The quotient identities dene the relationship among the trigonometric functions. Example 1 Graphing the Equations of an Identity Graph both sides of the identity cot θ = cot θ and y = . Solution See Figure 4. 1 tan θ http://https://legacy.cnx.org/content/m49393/1.5/ 1 tan θ . In other words, on the graphing calculator, graph y = OpenStax-CNX module: m49393 8 Figure 4 Analysis We see only one graph because both expressions generate the same image. One is on top of the other. This is a good way to prove any identity. If both expressions give the same graph, then they must be identities. How To: Given a trigonometric identity, verify that it is true. 1.Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build. 2.Look for opportunities to factor expressions, square a binomial, or add fractions. 3.Noting which functions are in the nal expression, look for opportunities to use the identities and make the proper substitutions. 4.If these steps do not yield the desired result, try converting all terms to sines and cosines. http://https://legacy.cnx.org/content/m49393/1.5/ OpenStax-CNX module: m49393 9 Example 2 Verifying a Trigonometric Identity Verify tan θcos θ = sin θ. Solution We will start on the left side, as it is the more complicated side: tan θcos θ = sin θ cos θ cos θ = sin θ )cos θ )cos θ (23) = sin θ Analysis This identity was fairly simple to verify, as it only required writing tan θ in terms of sin θ and cos θ. Try It: Exercise 3 (Solution on p. 18.) Verify the identity csc θcos θtan θ = 1. Example 3 Verifying the Equivalency Using the Even-Odd Identities Verify the following equivalency using the even-odd identities: (24) (1 + sin x) [1 + sin (−x)] = cos2 x Solution Working on the left side of the equation, we have (1 + sin x) [1 + sin (−x)] = (1 + sin x) (1 − sin x) 2 = 1 − sin x = cos2 x Example 4 Verifying a Trigonometric Identity Involving sec2 θ Verify the identity = sin θ Solution As the left side is more complicated, let's begin there. sec2 θ−1 sec2 θ 2 http://https://legacy.cnx.org/content/m49393/1.5/ Since sin(−x)= − sin x Dierence of squares cos x = 1 − sin x 2 2 (25) OpenStax-CNX module: m49393 sec2 θ−1 sec2 θ 10 = sec θ = tan θ + 1 (tan2 θ+1)−1 2 2 sec2 θ 2 θ = tan sec2 θ 2 1 sec2 θ 2 = tan θ = tan2 θ cos θ 2 sin θ cos2 θ = cos 2θ 2 = sin 2θ )cos2 θ cos2 θ = tan θ = 2 (26) 1 sec2 θ sin2 θ cos2 θ )cos θ 2 = sin θ There is more than one way to verify an identity. Here is another possibility. Again, we can start with the left side. sec2 θ−1 sec2 θ = sec2 θ sec2 θ − 1 sec2 θ (27) = 1 − cos2 θ = sin2 θ Analysis In the rst method, we used the identity sec θ = tan θ + 1 and continued to simplify. In the second method, we split the fraction, putting both terms in the numerator over the common denominator. This problem illustrates that there are multiple ways we can verify an identity. Employing some creativity can sometimes simplify a procedure. As long as the substitutions are correct, the answer will be the same. 2 2 Try It: Exercise 6 Show that (Solution on p. 18.) cot θ csc θ = cos θ. Example 5 Creating and Verifying an Identity Create an identity for the expression 2tan θsec θ by rewriting strictly in terms of sine. Solution There are a number of ways to begin, but here we will use the quotient and reciprocal identities to rewrite the expression: 2 tan θsec θ = 2 = Thus, = 1 sin θ cos θ cos θ 2 sin θ cos2 θ 2 sin θ 1−sin2 θ Substitute 1 − sin θ for cos 2tan θsec θ = http://https://legacy.cnx.org/content/m49393/1.5/ 2 2 sin θ 1 − sin2 θ (28) 2 θ (29) OpenStax-CNX module: m49393 11 Example 6 Verifying an Identity Using Algebra and Even/Odd Identities Verify the identity: (30) sin2 (−θ) − cos2 (−θ) = cos θ − sin θ sin (−θ) − cos (−θ) Solution Let's start with the left side and simplify: sin2 (−θ)−cos2 (−θ) sin(−θ)−cos(−θ) = [sin(−θ)]2 −[cos(−θ)]2 sin(−θ)−cos(−θ) θ)2 −(cos θ)2 = (−sin −sin θ−cos θ θ)2 −(cos θ)2 = (sin −sin θ−cos θ θ)(sin θ+cos θ) = (sin θ−cos −(sin θ+cos θ) (sin θ−cos θ)()sin θ+cos θ ) = and cos (−x) = cos x Dierence of squares (31) sin (−x) = −sin x −()sin θ+cos θ ) = cos θ − sin θ Try It: Exercise 9 Verify the identity (Solution on p. 18.) sin2 θ−1 tan θsin θ−tan θ = sin θ+1 tan θ . Example 7 Verifying an Identity Involving Cosines and Cotangents Verify the identity: 1 − cos x 1 + cot x = 1. Solution We will work on the left side of the equation. 2 1 − cos2 x 2 2 x 1 + cot2 x = 1 − cos2 x 1 + cos 2 sin x sin2 x cos2 x 2 = 1 − cos x sin2 x + sin2 x 2 x+cos2 x = 1 − cos2 x sin sin 2x 1 2 = sin x sin2 x =1 http://https://legacy.cnx.org/content/m49393/1.5/ Find the common denominator. (32) OpenStax-CNX module: m49393 12 2 Using Algebra to Simplify Trigonometric Expressions We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the dierence of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations. For example, the equation (sin x + 1) (sin x − 1) = 0 resembles the equation (x + 1) (x − 1) = 0, which uses the factored form of the dierence of squares. Using algebra makes nding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations. Another example is the dierence of squares formula, a − b = (a − b) (a + b) , which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve. 2 2 Example 8 Writing the Trigonometric Expression as an Algebraic Expression Write the following trigonometric expression as an algebraic expression: 2cos θ + cos θ − 1. Solution Notice that the pattern displayed has the same form as a standard quadratic expression, ax bx + c. Letting cos θ = x, we can rewrite the expression as follows: 2 2 + 2x2 + x − 1 This expression can be factored as (2x + 1) (x − 1) . If it were set equal to zero and we wanted to solve the equation, we would use the zero factor property and solve each factor for x. At this point, we would replace x with cos θ and solve for θ. (33) Example 9 Rewriting a Trigonometric Expression Using the Dierence of Squares Rewrite the trigonometric expression: 4 cos θ − 1. Solution Notice that both the coecient and the trigonometric expression in the rst term are squared, and the square of the number 1 is 1. This is the dierence of squares. Thus, 2 2 4 cos2 θ − 1 = (2 cos θ) − 1 = (2 cos θ − 1) (2 cos θ + 1) Analysis If this expression were written in the form of an equation set equal to zero, we could solve each factor using the zero factor property. We could also use substitution like we did in the previous problem and let cos θ = x, rewrite the expression as 4x − 1, and factor (2x − 1) (2x + 1) . Then replace x with cos θ and solve for the angle. 2 http://https://legacy.cnx.org/content/m49393/1.5/ (34) OpenStax-CNX module: m49393 13 Try It: Exercise 13 Rewrite the trigonometric expression: 25 − 9 sin (Solution on p. 18.) 2 θ. Example 10 Simplify by Rewriting and Using Substitution Simplify the expression by rewriting and using identities: (35) csc2 θ − cot2 θ Solution We can start with the Pythagorean identity. (36) 1 + cot2 θ = csc2 θ Now we can simplify by substituting 1 + cot θ for csc θ. We have 2 2 (37) csc2 θ − cot2 θ = 1 + cot2 θ − cot2 θ =1 Try It: Exercise 15 (Solution on p. 18.) = Use algebraic techniques to verify the identity: (Hint: Multiply the numerator and denominator on the left side by 1 − sin θ. cos θ 1+sin θ 1−sin θ cos θ . Access these online resources for additional instruction and practice with the fundamental trigonometric identities. • Fundamental Trigonometric Identities • Verifying Trigonometric Identities Media: 1 2 1 2 http://openstaxcollege.org/l/funtrigiden http://openstaxcollege.org/l/verifytrigiden http://https://legacy.cnx.org/content/m49393/1.5/ OpenStax-CNX module: m49393 14 3 Key Equations Pythagorean identities sin2 θ + cos2 θ = 1 1 + cot2 θ = csc2 θ 1 + tan2 θ = sec2 θ tan (−θ) = −tan θ cot (−θ) = −cot θ Even-odd identities sin (−θ) = −sin θ csc (−θ) = −csc θ cos (−θ) = cos θ sec (−θ) = sec θ 1 csc θ cos θ = sec1 θ tan θ = cot1 θ csc θ = sin1 θ sec θ = cos1 θ 1 cot θ = tan θ sin θ = Reciprocal identities Quotient identities tan θ = cot θ = sin θ cos θ cos θ sin θ Table 5 4 Key Concepts • • • • • • • There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem. Graphing both sides of an identity will verify it. See Example 1. Simplifying one side of the equation to equal the other side is another method for verifying an identity. See Example 2 and Example 3. The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation. See Example 4. We can create an identity by simplifying an expression and then verifying it. See Example 5. Verifying an identity may involve algebra with the fundamental identities. See Example 6 and Example 7. Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics. See Example 8, Example 9, and Example 10. http://https://legacy.cnx.org/content/m49393/1.5/ OpenStax-CNX module: m49393 15 5 Section Exercises 5.1 Verbal Exercise 16 (Solution on p. 18.) We know g (x) = cos x is an even function, and f (x) = sin x and h (x) = tan x are odd functions. What about G (x) = cos x, F (x) = sin x, and H (x) = tan x? Are they even, odd, or neither? Why? Exercise 17 Examine the graph of f (x) = sec x on the interval [−π, π] . How can we tell whether the function is even or odd by only observing the graph of f (x) = sec x? Exercise 18 (Solution on p. 18.) After examining the reciprocal identity for sec t, explain why the function is undened at certain points. Exercise 19 All of the Pythagorean identities are related. Describe how to manipulate the equations to get from sin t + cos t = 1 to the other forms. 2 2 2 2 2 5.2 Algebraic For the following exercises, use the fundamental identities to fully simplify the expression. Exercise 20 (Solution on p. 18.) sin x cos x sec x Exercise 21 sin (−x) cos (−x) csc (−x) Exercise 22 (Solution on p. 18.) tan xsin x + sec xcos2 x Exercise 23 csc x + cos xcot (−x) Exercise 24 (Solution on p. 18.) cot t+tan t sec(−t) Exercise 25 3 sin3 t csc t + cos2 t + 2 cos (−t) cos t Exercise 26 (Solution on p. 18.) −tan (−x) cot (−x) Exercise 27 −sin(−x)cos x sec x csc x tan x cot x Exercise 28 1+tan2 θ csc2 θ Exercise 29 tan x csc2 x + tan x sec2 x Exercise 30 1−cos2 x tan2 x (Solution on p. 18.) + sin2 θ + 1 sec2 θ 1+tan x 1+cot x − 1 cos2 x + 2 sin2 x (Solution on p. 18.) For the following exercises, simplify the rst trigonometric expression by writing the simplied form in terms of the second expression. http://https://legacy.cnx.org/content/m49393/1.5/ OpenStax-CNX module: m49393 16 Exercise 31 tan x+cot x ; csc x cos x sec x+csc x 1+tan x ; sin x Exercise 32 (Solution on p. 18.) Exercise 33 cos x 1+sin x + tan x; cos x Exercise 34 1 sin xcos x (Solution on p. 19.) − cot x; cot x Exercise 35 1 1−cos x − cos x 1+cos x ; csc x Exercise 36 (Solution on p. 19.) (sec x + csc x) (sin x + cos x) − 2 − cot x; tan x Exercise 37 1 csc x−sin x ; sec x Exercise 38 1−sin x 1+sin x − 1+sin x 1−sin x Exercise 39 and tan x ; sec x and tan x (Solution on p. 19.) tan x; sec x Exercise 40 (Solution on p. 19.) sec x; cot x Exercise 41 sec x; sin x Exercise 42 (Solution on p. 19.) cot x; sin x Exercise 43 cot x; csc x For the following exercises, verify the identity. Exercise 44 (Solution on p. 19.) cos x − cos3 x = cos x sin2 x Exercise 45 cos x (tan x − sec (−x)) = sin x − 1 Exercise 46 1+sin2 x cos2 x = (Solution on p. 19.) 1 cos2 x Exercise 47 + sin2 x cos2 x = 1 + 2 tan2 x 2 (sin x + cos x) = 1 + 2 sin xcos x Exercise 48 cos2 x − tan2 x = 2 − sin2 x − sec2 x http://https://legacy.cnx.org/content/m49393/1.5/ (Solution on p. 19.) OpenStax-CNX module: m49393 17 5.3 Extensions For the following exercises, prove or disprove the identity. Exercise 49 1 1+cos x − 1 1−cos(−x) = −2 cot x csc x Exercise 50 (Solution on p. 19.) csc2 x 1 + sin2 x = cot2 x Exercise 51 sec2 (−x)−tan2 x tan x Exercise 52 tan x sec x sin (−x) 2+2 tan x 2+2 cot x − 2 sin2 x = cos 2x (Solution on p. 19.) = cos2 x Exercise 53 sec(−x) tan x+cot x = −sin (−x) Exercise 54 1+sin x cos x = (Solution on p. 19.) cos x 1+sin(−x) For the following exercises, determine whether the identity is true or false. If false, nd an appropriate equivalent expression. Exercise 55 cos2 θ−sin2 θ 1−tan2 θ = sin2 θ Exercise 56 3 sin2 θ + 4 cos2 θ = 3 + cos2 θ Exercise 57 sec θ+tan θ cot θ+cos θ = sec2 θ http://https://legacy.cnx.org/content/m49393/1.5/ (Solution on p. 19.) OpenStax-CNX module: m49393 18 Solutions to Exercises in this Module Solution to Exercise (p. 9) 1 sin θ sin θ cos θ cos θ cos θ sin θ sin θ cos θ sin θcos θ sin θcos θ csc θcos θtan θ = = = (38) =1 Solution to Exercise (p. 10) cot θ csc θ = cos θ sin θ 1 sin θ = cos θ sin θ · (39) sin θ 1 = cos θ Solution to Exercise (p. 11) sin2 θ−1 tan θsin θ−tan θ = (sin θ+1)(sin θ−1) tan θ(sin θ−1) sin θ+1 tan θ = Solution to Exercise (p. 13) This is a dierence of squares formula: 25 − 9 sin 2 Solution to Exercise (p. 13) cos θ 1+sin θ 1−sin θ 1−sin θ θ = (5 − 3 sin θ) (5 + 3 sin θ) . = cos θ(1−sin θ) 1−sin2 θ θ) = cos θ(1−sin cos2 θ θ = 1−sin cos θ (40) Solution to Exercise (p. 15) All three functions, F, G, and H, are even. This is because F (−x) = sin (−x) sin (−x) = (−sin x) (−sin x) = sin x = F (x) , G (−x) = cos (−x) cos (−x) = cos xcos x = cos x = G (x) and H (−x) = tan (−x) tan (−x) = (−tan x) (−tan x) = tan x = H (x) . Solution to Exercise (p. 15) When cos t = 0, then sec t = , which is undened. 2 2 2 1 0 Solution to Exercise (p. 15) sin x Solution to Exercise (p. 15) sec x Solution to Exercise (p. 15) csc t Solution to Exercise (p. 15) −1 Solution to Exercise (p. 15) sec2 x Solution to Exercise (p. 15) sin2 x + 1 http://https://legacy.cnx.org/content/m49393/1.5/ OpenStax-CNX module: m49393 19 Solution to Exercise (p. 16) 1 sin x Solution to Exercise (p. 16) 1 cot x Solution to Exercise (p. 16) tan x Solution to Exercise (p. 16) −4sec xtan x Solution to Exercise (p. 16) q ± 1 cot2 x ± 1−sin2 x sin x +1 Solution to Exercise (p. 16) √ Solution to Exercise (p. 16) Answers will vary. Sample proof: cos x − cos3 x = cos x 1 − cos2 x = cos xsin2 x Solution to Exercise (p. 16) Answers will vary. Sample proof: 1+sin2 x cos2 x = 1 cos2 x + sin2 x cos2 x = sec2 x + tan2 x = tan2 x + 1 + tan2 x = 1 + 2tan2 x Solution to Exercise (p. 16) Answers will vary. Sample proof: cos2 x − tan2 x = 1 − sin2 x − sec2 x − 1 = 1 − sin2 x − sec2 x + 1 = 2 − sin2 x − sec2 x Solution to Exercise (p. 17) False False Solution to Exercise (p. 17) Proved with negative and Pythagorean identities Solution to Exercise (p. 17) True 3 sin θ + 4 cos θ = 3 sin θ + 3 cos θ + cos θ = 3 Solution to Exercise (p. 17) 2 2 2 2 2 sin2 θ + cos2 θ + cos2 θ = 3 + cos2 θ Glossary Denition 1: even-odd identities set of equations involving trigonometric functions such that if f (−x) = −f (x) ,the identity is odd, and if f (−x) = f (x) ,the identity is even Denition 2: Pythagorean identities set of equations involving trigonometric functions based on the right triangle properties Denition 3: quotient identities pair of identities based on the fact that tangent is the ratio of sine and cosine, and cotangent is the ratio of cosine and sine Denition 4: reciprocal identities set of equations involving the reciprocals of basic trigonometric denitions http://https://legacy.cnx.org/content/m49393/1.5/