770 Foundations of Trigonometry 0. Trigonometric Identities In Section 0.3, we saw the utility of the Pythagorean Identities in Theorem 0.8 along with the Quotient and Reciprocal Identities in Theorem 0.6. Not only did these identities help us compute the values of the circular functions for angles, they were also useful in simplifying expressions involving the circular functions. In this section, we introduce several collections of identities which have uses in this course and beyond. Our first set of identities is the Even / Odd identities. Theorem 0.. Even / Odd Identities: For all applicable angles, cos( cos( sec( sec( sin( sin( csc( csc( tan( tan( cot( cot( In light of the Quotient and Reciprocal Identities, Theorem 0.6, it suffices to show cos( cos( and sin( sin(. The remaining four circular functions can be expressed in terms of cos( and sin( so the proofs of their Even / Odd Identities are left as exercises. Consider an angle plotted in standard position. Let 0 be the angle coterminal with with 0 0 < π. (We can construct the angle 0 by rotating counter-clockwise from the positive x-axis to the terminal side of as pictured below. Since and 0 are coterminal, cos( cos( 0 and sin( sin( 0. y y 0 0 P (cos( 0, sin( 0 x Q(cos( 0, sin( 0 x 0 We now consider the angles and 0. Since is coterminal with 0, there is some integer k so that 0 + π k. Therefore, 0 π k 0 + π ( k. Since k is an integer, so is ( k, which means is coterminal with 0. Hence, cos( cos( 0 and sin( sin( 0. Let P and Q denote the points on the terminal sides of 0 and 0, respectively, which lie on the Unit Circle. By definition, the coordinates of P are (cos( 0, sin( 0 and the coordinates of Q are (cos( 0, sin( 0. Since 0 and 0 sweep out congruent central sectors of the Unit Circle, it As mentioned at the end of Section 0., properties of the circular functions when thought of as functions of angles in radian measure hold equally well if we view these functions as functions of real numbers. Not surprisingly, the Even / Odd properties of the circular functions are so named because they identify cosine and secant as even functions, while the remaining four circular functions are odd. (See Section.6.
0. Trigonometric Identities 77 follows that the points P and Q are symmetric about the x-axis. Thus, cos( 0 cos( 0 and sin( 0 sin( 0. Since the cosines and sines of 0 and 0 are the same as those for and, respectively, we get cos( cos( and sin( sin(, as required. The Even / Odd Identities are readily demonstrated using any of the common angles noted in Section 0.. Their true utility, however, lies not in computation, but in simplifying expressions involving the circular functions. In fact, our next batch of identities makes heavy use of the Even / Odd Identities. Theorem 0.3. Sum and Difference Identities for Cosine: For all angles α and β, cos(α + β cos(α cos(β sin(α sin(β cos(α β cos(α cos(β + sin(α sin(β We first prove the result for differences. As in the proof of the Even / Odd Identities, we can reduce the proof for general angles α and β to angles α 0 and β 0, coterminal with α and β, respectively, each of which measure between 0 and π radians. Since α and α 0 are coterminal, as are β and β 0, it follows that α β is coterminal with α 0 β 0. Consider the case below where α 0 β 0. y y P (cos(α 0, sin(α 0 α 0 β 0 Q(cos(β 0, sin(β 0 A(cos(α 0 β 0, sin(α 0 β 0 α 0 β 0 α 0 β 0 O x O B(, 0 x Since the angles P OQ and AOB are congruent, the distance between P and Q is equal to the distance between A and B. The distance formula, Equation., yields (cos(α0 cos(β 0 + (sin(α 0 sin(β 0 (cos(α 0 β 0 + (sin(α 0 β 0 0 Squaring both sides, we expand the left hand side of this equation as (cos(α 0 cos(β 0 + (sin(α 0 sin(β 0 cos (α 0 cos(α 0 cos(β 0 + cos (β 0 + sin (α 0 sin(α 0 sin(β 0 + sin (β 0 cos (α 0 + sin (α 0 + cos (β 0 + sin (β 0 cos(α 0 cos(β 0 sin(α 0 sin(β 0 In the picture we ve drawn, the triangles P OQ and AOB are congruent, which is even better. However, α 0 β 0 could be 0 or it could be π, neither of which makes a triangle. It could also be larger than π, which makes a triangle, just not the one we ve drawn. You should think about those three cases.
77 Foundations of Trigonometry From the Pythagorean Identities, cos (α 0 + sin (α 0 and cos (β 0 + sin (β 0, so (cos(α 0 cos(β 0 + (sin(α 0 sin(β 0 cos(α 0 cos(β 0 sin(α 0 sin(β 0 Turning our attention to the right hand side of our equation, we find (cos(α 0 β 0 + (sin(α 0 β 0 0 cos (α 0 β 0 cos(α 0 β 0 + + sin (α 0 β 0 + cos (α 0 β 0 + sin (α 0 β 0 cos(α 0 β 0 Once again, we simplify cos (α 0 β 0 + sin (α 0 β 0, so that (cos(α 0 β 0 + (sin(α 0 β 0 0 cos(α 0 β 0 Putting it all together, we get cos(α 0 cos(β 0 sin(α 0 sin(β 0 cos(α 0 β 0, which simplifies to: cos(α 0 β 0 cos(α 0 cos(β 0 + sin(α 0 sin(β 0. Since α and α 0, β and β 0 and α β and α 0 β 0 are all coterminal pairs of angles, we have cos(α β cos(α cos(β + sin(α sin(β. For the case where α 0 β 0, we can apply the above argument to the angle β 0 α 0 to obtain the identity cos(β 0 α 0 cos(β 0 cos(α 0 + sin(β 0 sin(α 0. Applying the Even Identity of cosine, we get cos(β 0 α 0 cos( (α 0 β 0 cos(α 0 β 0, and we get the identity in this case, too. To get the sum identity for cosine, we use the difference formula along with the Even/Odd Identities cos(α + β cos(α ( β cos(α cos( β + sin(α sin( β cos(α cos(β sin(α sin(β We put these newfound identities to good use in the following example. Example 0.... Find the exact value of cos (.. Verify the identity: cos sin(. Solution.. In order to use Theorem 0.3 to find cos (, we need to write as a sum or difference of angles whose cosines and sines we know. One way to do so is to write 30. cos ( cos ( 30 cos ( cos (30 + sin ( sin (30 ( ( ( ( 3 + 6 +
0. Trigonometric Identities 773. In a straightforward application of Theorem 0.3, we find cos cos cos ( + sin sin ( (0 (cos( + ( (sin( sin( The identity verified in Example 0.., namely, cos sin(, is the first of the celebrated cofunction identities. These identities were first hinted at in Exercise 7 in Section 0.. From sin( cos, we get: [ π ] sin cos cos(, which says, in words, that the co sine of an angle is the sine of its co mplement. Now that these identities have been established for cosine and sine, the remaining circular functions follow suit. The remaining proofs are left as exercises. Theorem 0.. Cofunction Identities: For all applicable angles, cos sin( sec csc( tan cot( sin cos( csc sec( cot tan( With the Cofunction Identities in place, we are now in the position to derive the sum and difference formulas for sine. To derive the sum formula for sine, we convert to cosines using a cofunction identity, then expand using the difference formula for cosine sin(α + β cos (α + β ([ π ] cos α β cos α cos(β + sin α sin(β sin(α cos(β + cos(α sin(β We can derive the difference formula for sine by rewriting sin(α β as sin(α + ( β and using the sum formula and the Even / Odd Identities. Again, we leave the details to the reader. Theorem 0.. Sum and Difference Identities for Sine: For all angles α and β, sin(α + β sin(α cos(β + cos(α sin(β sin(α β sin(α cos(β cos(α sin(β
77 Foundations of Trigonometry Example 0.... Find the exact value of sin ( 9π. If α is a Quadrant II angle with sin(α 3, and β is a Quadrant III angle with tan(β, find sin(α β. 3. Derive a formula for tan(α + β in terms of tan(α and tan(β. Solution.. As in Example 0.., we need to write the angle 9π as a sum or difference of common angles. The denominator of suggests a combination of angles with denominators 3 and. One such combination is 9π π 3 + π. Applying Theorem 0., we get ( ( 9π π sin sin 3 + π ( π ( π sin cos + cos sin 3 3 ( ( ( 3 + ( 6. In order to find sin(α β using Theorem 0., we need to find cos(α and both cos(β and sin(β. To find cos(α, we use the Pythagorean Identity cos (α + sin (α. Since sin(α 3, we have cos (α + ( 3, or cos(α ± 3. Since α is a Quadrant II angle, cos(α 3. We now set about finding cos(β and sin(β. We have several ways to proceed, but the Pythagorean Identity + tan (β sec (β is a quick way to get sec(β, and hence, cos(β. With tan(β, we get + sec (β so that sec(β ±. Since β is a Quadrant III angle, we choose sec(β so cos(β sec(β. We now need to determine sin(β. We could use The Pythagorean Identity cos (β + sin (β, but we opt instead to use a quotient identity. From tan(β sin(β cos(β, we have sin(β tan(β cos(β ( so we get sin(β (. We now have all the pieces needed to find sin(α β: sin(α β sin(α cos(β cos(α sin(β ( ( ( ( 3 3 9 6
0. Trigonometric Identities 77 3. We can start expanding tan(α + β using a quotient identity and our sum formulas tan(α + β sin(α + β cos(α + β sin(α cos(β + cos(α sin(β cos(α cos(β sin(α sin(β Since tan(α sin(α sin(β cos(α and tan(β cos(β, it looks as though if we divide both numerator and denominator by cos(α cos(β we will have what we want tan(α + β sin(α cos(β + cos(α sin(β cos(α cos(β sin(α sin(β cos(α cos(β cos(α cos(β sin(α cos(β cos(α sin(β + cos(α cos(β cos(α cos(β cos(α cos(β sin(α sin(β cos(α cos(β cos(α cos(β sin(α cos(β cos(α cos(β + cos(α sin(β cos(α cos(β cos(α cos(β sin(α sin(β cos(α cos(β cos(α cos(β tan(α + tan(β tan(α tan(β Naturally, this formula is limited to those cases where all of the tangents are defined. The formula developed in Exercise 0.. for tan(α+β can be used to find a formula for tan(α β by rewriting the difference as a sum, tan(α+( β, and the reader is encouraged to fill in the details. Below we summarize all of the sum and difference formulas for cosine, sine and tangent. Theorem 0.6. Sum and Difference Identities: For all applicable angles α and β, cos(α ± β cos(α cos(β sin(α sin(β sin(α ± β sin(α cos(β ± cos(α sin(β tan(α ± β tan(α ± tan(β tan(α tan(β In the statement of Theorem 0.6, we have combined the cases for the sum + and difference of angles into one formula. The convention here is that if you want the formula for the sum + of
776 Foundations of Trigonometry two angles, you use the top sign in the formula; for the difference,, use the bottom sign. For example, tan(α tan(β tan(α β + tan(α tan(β If we specialize the sum formulas in Theorem 0.6 to the case when α β, we obtain the following Double Angle Identities. Theorem 0.7. Double Angle Identities: For all applicable angles, cos ( sin ( cos( cos ( sin ( sin( sin( cos( tan( tan( tan ( The three different forms for cos( can be explained by our ability to exchange squares of cosine and sine via the Pythagorean Identity cos ( + sin ( and we leave the details to the reader. It is interesting to note that to determine the value of cos(, only one piece of information is required: either cos( or sin(. To determine sin(, however, it appears that we must know both sin( and cos(. In the next example, we show how we can find sin( knowing just one piece of information, namely tan(. Example 0..3.. Suppose P ( 3, lies on the terminal side of when is plotted in standard position. Find cos( and sin( and determine the quadrant in which the terminal side of the angle lies when it is plotted in standard position.. If sin( x for π π, find an expression for sin( in terms of x. 3. Verify the identity: sin( tan( + tan (.. Express cos(3 as a polynomial in terms of cos(. Solution.. Using Theorem 0.3 from Section 0. with x 3 and y, we find r x + y. Hence, cos( 3 and sin(. Applying Theorem 0.7, we get cos( cos ( sin ( ( 3 ( 7, and sin( sin( cos( ( ( 3. Since both cosine and sine of are negative, the terminal side of, when plotted in standard position, lies in Quadrant III.
0. Trigonometric Identities 777. If your first reaction to sin( x is No it s not, cos( x! then you have indeed learned something, and we take comfort in that. However, context is everything. Here, x is just a variable - it does not necessarily represent the x-coordinate of the point on The Unit Circle which lies on the terminal side of, assuming is drawn in standard position. Here, x represents the quantity sin(, and what we wish to know is how to express sin( in terms of x. We will see more of this kind of thing in Section 0.6, and, as usual, this is something we need for Calculus. Since sin( sin( cos(, we need to write cos( in terms of x to finish the problem. We substitute x sin( into the Pythagorean Identity, cos ( + sin (, to get cos ( + x, or cos( ± x. Since π π, cos( 0, and thus cos( x. Our final answer is sin( sin( cos( x x. 3. We start with the right hand side of the identity and note that + tan ( sec (. From this point, we use the Reciprocal and Quotient Identities to rewrite tan( and sec( in terms of cos( and sin(: ( sin( tan( + tan tan( ( sin( ( sec ( cos ( cos( cos( cos ( ( sin( cos( cos( cos( sin( cos( sin(. In Theorem 0.7, one of the formulas for cos(, namely cos( cos (, expresses cos( as a polynomial in terms of cos(. We are now asked to find such an identity for cos(3. Using the sum formula for cosine, we begin with cos(3 cos( + cos( cos( sin( sin( Our ultimate goal is to express the right hand side in terms of cos( only. We substitute cos( cos ( and sin( sin( cos( which yields cos(3 cos( cos( sin( sin( ( cos ( cos( ( sin( cos( sin( cos 3 ( cos( sin ( cos( Finally, we exchange sin ( for cos ( courtesy of the Pythagorean Identity, and get and we are done. cos(3 cos 3 ( cos( sin ( cos( cos 3 ( cos( ( cos ( cos( cos 3 ( cos( cos( + cos 3 ( cos 3 ( 3 cos(
778 Foundations of Trigonometry In the last problem in Example 0..3, we saw how we could rewrite cos(3 as sums of powers of cos(. In Calculus, we have occasion to do the reverse; that is, reduce the power of cosine and sine. Solving the identity cos( cos ( for cos ( and the identity cos( sin ( for sin ( results in the aptly-named Power Reduction formulas below. Theorem 0.8. Power Reduction Formulas: For all angles, cos ( + cos( sin ( cos( Example 0... Rewrite sin ( cos ( as a sum and difference of cosines to the first power. Solution. We begin with a straightforward application of Theorem 0.8 sin ( cos ( ( ( cos( + cos( ( cos ( cos ( Next, we apply the power reduction formula to cos ( to finish the reduction sin ( cos ( cos ( ( + cos(( 8 8 cos( 8 8 cos( Another application of the Power Reduction Formulas is the Half Angle Formulas. To start, we apply the Power Reduction Formula to cos ( cos ( + cos ( ( + cos(. We can obtain a formula for cos ( by extracting square roots. In a similar fashion, we may obtain a half angle formula for sine, and by using a quotient formula, obtain a half angle formula for tangent. We summarize these formulas below.
0. Trigonometric Identities 779 Theorem 0.9. Half Angle Formulas: For all applicable angles, ( + cos( cos ± ( cos( sin ± ( cos( tan ± + cos( where the choice of ± depends on the quadrant in which the terminal side of lies. Example 0.... Use a half angle formula to find the exact value of cos (.. Suppose π 0 with cos( 3. Find sin (. 3. Use the identity given in number 3 of Example 0..3 to derive the identity ( tan sin( + cos( Solution.. To use the half angle formula, we note that 30 and since is a Quadrant I angle, its cosine is positive. Thus we have + cos (30 cos ( + 3 + + 3 + 3 + 3 Back in Example 0.., we found cos ( by using the difference formula for cosine. In that case, we determined cos ( 6+. The reader is encouraged to prove that these two expressions are equal.. If π 0, then π 0, which means sin ( < 0. Theorem 0.9 gives ( ( cos ( 3 sin + 3 8 0
780 Foundations of Trigonometry 3. Instead of our usual approach to verifying identities, namely starting with one side of the equation and trying to transform it into the other, we will start with the identity we proved in number 3 of Example 0..3 and manipulate it into the identity we are asked to prove. The identity we are asked to start with is sin( tan(. If we are to use this to derive an identity for tan ( +tan (, it seems reasonable to proceed by replacing each occurrence of with sin ( ( tan ( + tan ( tan ( sin( + tan ( We now have the sin( we need, but we somehow need to get a factor of + cos( involved. To get cosines involved, recall that + tan ( sec (. We continue to manipulate our given identity by converting secants to cosines and using a power reduction formula tan ( sin( + tan ( sin( tan ( sec ( sin( tan ( cos ( sin( tan ( ( + cos ( ( sin( tan ( ( + cos( ( sin( tan + cos( Our next batch of identities, the Product to Sum Formulas, 3 are easily verified by expanding each of the right hand sides in accordance with Theorem 0.6 and as you should expect by now we leave the details as exercises. They are of particular use in Calculus, and we list them here for reference. Theorem 0.0. Product to Sum Formulas: For all angles α and β, cos(α cos(β [cos(α β + cos(α + β] sin(α sin(β [cos(α β cos(α + β] sin(α cos(β [sin(α β + sin(α + β] 3 These are also known as the Prosthaphaeresis Formulas and have a rich history. The authors recommend that you conduct some research on them as your schedule allows.
0. Trigonometric Identities 78 Related to the Product to Sum Formulas are the Sum to Product Formulas, which we will have need of in Section 0.7. These are easily verified using the Product to Sum Formulas, and as such, their proofs are left as exercises. Theorem 0.. Sum to Product Formulas: For all angles α and β, ( ( α + β α β cos(α + cos(β cos cos ( ( α + β α β cos(α cos(β sin sin ( ( α ± β α β sin(α ± sin(β sin cos Example 0..6.. Write cos( cos(6 as a sum.. Write sin( sin(3 as a product. Solution.. Identifying α and β 6, we find cos( cos(6 [cos( 6 + cos( + 6] cos( + cos(8 cos( + cos(8, where the last equality is courtesy of the even identity for cosine, cos( cos(.. Identifying α and β 3 yields ( 3 sin( sin(3 sin cos sin ( cos ( sin ( cos (, ( + 3 where the last equality is courtesy of the odd identity for sine, sin( sin(. The reader is reminded that all of the identities presented in this section which regard the circular functions as functions of angles (in radian measure apply equally well to the circular (trigonometric functions regarded as functions of real numbers. In Exercises 38-3 in Section 0., we see how some of these identities manifest themselves geometrically as we study the graphs of the these functions. In the upcoming Exercises, however, you need to do all of your work analytically without graphs.
78 Foundations of Trigonometry 0.. Exercises In Exercises - 6, use the Even / Odd Identities to verify the identity. Assume all quantities are defined.. sin(3π sin( 3π. cos ( π ( t cos t + π 3. tan( t + tan(t. csc( csc( +. sec( 6t sec(6t 6. cot(9 7 cot(7 9 In Exercises 7 -, use the Sum and Difference Identities to find the exact value. You may have need of the Quotient, Reciprocal or Even / Odd Identities as well. 7. cos(7 8. sec(6 9. sin(0 0. csc(9. cot(. tan(37 ( ( 3π π 3. cos. sin ( ( 7π 7π 6. cos 7. tan ( ( π π 9. cot 0. csc. If α is a Quadrant IV angle with cos(α, and sin(β. tan ( 3π 8. sin (. sec π 0 0, where π (a cos(α + β (b sin(α + β (c tan(α + β (d cos(α β (e sin(α β (f tan(α β < β < π, find 3. If csc(α 3, where 0 < α < π, and β is a Quadrant II angle with tan(β 7, find (a cos(α + β (b sin(α + β (c tan(α + β (d cos(α β (e sin(α β (f tan(α β. If sin(α 3, where 0 < α < π, and cos(β 3 where 3π < β < π, find (a sin(α + β (b cos(α β (c tan(α β
0. Trigonometric Identities 783. If sec(α 3, where π < α < π, and tan(β 7, where π < β < 3π, find (a csc(α β (b sec(α + β (c cot(α + β In Exercises 6-38, verify the identity. 6. cos( π cos( 7. sin(π sin( ( 8. tan + π cot( 9. sin(α + β + sin(α β sin(α cos(β 30. sin(α + β sin(α β cos(α sin(β 3. cos(α + β + cos(α β cos(α cos(β 3. cos(α + β cos(α β sin(α sin(β 33. sin(α + β + cot(α tan(β sin(α β cot(α tan(β 3. 36. 37. 38. cos(α + β tan(α tan(β cos(α β + tan(α tan(β sin(t + h sin(t h cos(t + h cos(t h tan(t + h tan(t h tan(α + β 3. tan(α β ( ( sin(h cos(h cos(t + sin(t h h ( cos(h cos(t h sin(t ( ( tan(h sec (t h tan(t tan(h ( sin(h h sin(α cos(α + sin(β cos(β sin(α cos(α sin(β cos(β In Exercises 39-8, use the Half Angle Formulas to find the exact value. You may have need of the Quotient, Reciprocal or Even / Odd Identities as well. 39. cos(7 (compare with Exercise 7 0. sin(0 (compare with Exercise 9. cos(67.. sin(7. ( 7π 3. tan(.. cos (compare with Exercise 6. sin (compare with Exercise 8 6. cos 8 7. sin ( π 8 8. tan ( 7π 8
78 Foundations of Trigonometry In Exercises 9-8, use the given information about to find the exact values of sin( ( sin 9. sin( 7 where 3π cos( ( cos tan( ( tan < < π 0. cos( 8 3 where 0 < < π. tan( where π < < 3π. csc( where π < < π 3. cos( 3 where 0 < < π. sin( where π < < 3π. cos( 3 where 3π < < π 6. sin( 3 where π < < π 7. sec( where 3π < < π 8. tan( where π < < π In Exercises 9-73, verify the identity. Assume all quantities are defined. 9. (cos( + sin( + sin( 60. (cos( sin( sin( 6. tan( tan( + tan( 6. csc( cot( + tan( 63. 8 sin ( cos( cos( + 3 6. 8 cos ( cos( + cos( + 3 6. sin(3 3 sin( sin 3 ( 66. sin( sin( cos 3 ( sin 3 ( cos( 67. 3 sin ( cos ( + cos( cos( cos(6 68. 3 sin ( cos ( cos( cos( + cos(6 69. cos( 8 cos ( 8 cos ( + 70. cos(8 8 cos 8 ( 6 cos 6 ( + 60 cos ( 3 cos ( + (HINT: Use the result to 69. 7. sec( 7. 73. cos( cos( + sin( + sin( cos( sin( cos( sin( + cos( + sin( cos( cos( cos( sin( cos( + sin( sin( cos(
0. Trigonometric Identities 78 In Exercises 7-79, write the given product as a sum. You may need to use an Even/Odd Identity. 7. cos(3 cos( 7. sin( sin(7 76. sin(9 cos( 77. cos( cos(6 78. sin(3 sin( 79. cos( sin(3 In Exercises 80-8, write the given sum as a product. You may need to use an Even/Odd or Cofunction Identity. 80. cos(3 + cos( 8. sin( sin(7 8. cos( cos(6 83. sin(9 sin( 8. sin( + cos( 8. cos( sin( 86. Suppose is a Quadrant I angle with sin( x. Verify the following formulas (a cos( x (b sin( x x (c cos( x 87. Discuss with your classmates how each of the formulas, if any, in Exercise 86 change if we change assume is a Quadrant II, III, or IV angle. 88. Suppose is a Quadrant I angle with tan( x. Verify the following formulas (a cos( x + (b sin( x x + (c sin( x x + (d cos( x x + 89. Discuss with your classmates how each of the formulas, if any, in Exercise 88 change if we change assume is a Quadrant II, III, or IV angle. 90. If sin( x for π < < π, find an expression for cos( in terms of x. 9. If tan( x 7 for π < < π, find an expression for sin( in terms of x. 9. If sec( x for 0 < < π, find an expression for ln sec( + tan( in terms of x. 93. Show that cos ( sin ( cos ( sin ( for all. 9. Let be a Quadrant III angle with cos(. Show that this is not enough information to ( determine the sign of sin by first assuming 3π < < 7π and then assuming π < < 3π ( and computing sin in both cases.
786 Foundations of Trigonometry 9. Without using your calculator, show that + 3 6 + 96. In part of Example 0..3, we wrote cos(3 as a polynomial in terms of cos(. In Exercise 69, we had you verify an identity which expresses cos( as a polynomial in terms of cos(. Can you find a polynomial in terms of cos( for cos(? cos(6? Can you find a pattern so that cos(n could be written as a polynomial in cosine for any natural number n? 97. In Exercise 6, we has you verify an identity which expresses sin(3 as a polynomial in terms of sin(. Can you do the same for sin(? What about for sin(? If not, what goes wrong? 98. Verify the Even / Odd Identities for tangent, secant, cosecant and cotangent. 99. Verify the Cofunction Identities for tangent, secant, cosecant and cotangent. 00. Verify the Difference Identities for sine and tangent. 0. Verify the Product to Sum Identities. 0. Verify the Sum to Product Identities.
0. Trigonometric Identities 787 0.. Answers 6 7. cos(7 6 + 9. sin(0 8. sec(6 6 + 6 0. csc(9 6 ( + 6. cot( 3. cos 3 3 + 3. tan(37 3 3 ( 3π 6 +. sin ( π 3 + 3 3 6 ( 3π. tan 3 3 3 + 3 ( 7π 6 3 6. cos ( 7π 7. tan + ( π 6 3 8. sin ( π 9. cot ( + ( π 3 0. csc 6 (. sec π 6. (a cos(α + β (c tan(α + β 7 0 (b sin(α + β 7 0 (d cos(α β (e sin(α β (f tan(α β 3. (a cos(α + β + 7 30 (c tan(α + β 8 + + 7 63 00 (b sin(α + β 8 30 (d cos(α β + 7 30 (e sin(α β 8 + 30. (a sin(α + β 6 6 (b cos(α β 33 6 (f tan(α β 8 + 7 63 + 00 (c tan(α β 6 33
788 Foundations of Trigonometry. (a csc(α β (b sec(α + β 7 (c cot(α + β 7 39. cos(7 3 0. sin(0 + 3. cos(67. 3. tan(. +. cos. sin(7. ( 7π 3 ( π 3. sin ( π + 7. sin 8 + 6. cos 8 8. tan ( 7π 8 + 9. sin( 336 6 sin ( 0 0. sin( 0 809 sin ( 06 06. sin( 0 69 sin ( 3 3 3. sin( 8 sin ( 8 + 3. sin( sin ( cos( 7 6 cos ( 7 0 cos( 809 cos ( 9 06 06 cos( 9 69 cos ( 3 3 cos( 7 8 cos ( 8 cos( 7 cos ( tan( 336 7 tan ( 7 tan( 0 tan ( 9 tan( 0 9 tan ( 3 tan( 7 tan ( 8 + 8 tan ( + tan( 7 tan (
0. Trigonometric Identities 789. sin( sin (. sin( 0 69 sin ( 6 6 6. sin( 0 69 sin ( 6 6 cos( 7 cos ( cos( 9 69 cos ( 6 6 cos( 9 69 cos ( 6 6 tan( 7 tan ( tan( 0 9 tan ( tan( 0 9 tan ( 7. sin( sin ( 0 0 0 cos( 3 cos ( 0 + 0 0 tan( 3 tan ( + tan ( 0 8. sin( sin ( 0 + 0 0 cos( 3 cos ( 0 0 0 tan( 3 tan ( + tan ( + 0 7. 77. cos( + cos(8 cos( + cos(8 80. cos( cos( 7. 78. cos( cos(9 cos( cos( ( 9 8. cos 83. cos( sin( 8. cos ( π ( sin 76. sin(8 + sin(0 sin( + sin( 79. ( ( 8. sin sin 8. ( sin π 90. x 9. x x + 9 9. ln x + x + 6 ln(