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# 7.2E: Exercises - Sum and Difference Identities

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### A: Evaluate sum and difference formulas from a given angle

Exercise $$\PageIndex{A}$$

$$\bigstar$$ Find the exact value.

 1. $$\cos \left ( \dfrac{7\pi }{4}- \dfrac{5\pi }{3} \right) \\[4pt]$$ 2. $$\cos \left ( \dfrac{11\pi }{6}- \dfrac{3\pi }{4} \right) \\[4pt]$$ 3. $$\cos \left ( \dfrac{\pi }{4}+ \dfrac{2\pi }{3} \right) \\[4pt]$$ 4. $$\cos \left ( \dfrac{5\pi }{4}+ \dfrac{\pi }{3} \right) \\[4pt]$$ 5. $$\sin \left ( \dfrac{5\pi }{4}- \dfrac{7\pi }{6} \right) \\[4pt]$$ 6. $$\sin \left ( \dfrac{5\pi }{3}- \dfrac{3\pi }{4} \right) \\[4pt]$$ 7. $$\sin \left ( \dfrac{\pi }{4}+ \dfrac{5\pi }{6} \right) \\[4pt]$$ 8. $$\sin \left ( \dfrac{7\pi }{4}+ \dfrac{\pi }{6} \right) \\[4pt]$$ 9. $$\tan \left ( \dfrac{11\pi }{6}- \dfrac{5\pi }{4} \right) \\[4pt]$$ 10. $$\tan \left ( \dfrac{7\pi }{4}- \dfrac{2\pi }{3} \right) \\[4pt]$$ 11. $$\tan \left ( \dfrac{\pi }{4}+ \dfrac{\pi }{6} \right) \\[4pt]$$ 12. $$\tan \left ( \dfrac{3\pi }{4}+ \dfrac{7\pi }{6} \right) \\[4pt]$$

$$\bigstar$$ Find the exact value.

 13. $$\cos \left (\dfrac{\pi }{12} \right) = \cos \left ( \dfrac{4\pi }{3}- \dfrac{5\pi }{4} \right) \\[4pt]$$ 14. $$\cos \left (\dfrac{7\pi }{12} \right) = \cos \left ( \dfrac{5\pi }{6}- \dfrac{\pi }{4} \right) \\[4pt]$$ 15. $$\sin \left (\dfrac{11\pi }{12} \right) = \sin \left ( \dfrac{3\pi }{4}+ \dfrac{\pi }{6} \right)$$ 16. $$\sin \left (\dfrac{5\pi }{12} \right) = \sin \left ( \dfrac{7\pi }{4}- \dfrac{4\pi }{3} \right) \\[4pt]$$ 17. $$\tan \left (\dfrac{19\pi }{12} \right) = \tan \left ( \dfrac{3\pi }{4}+ \dfrac{5\pi }{6} \right) \\[4pt]$$ 18. $$\tan \left (-\dfrac{\pi }{12} \right) = \tan \left ( \dfrac{7\pi }{6}- \dfrac{5\pi }{4} \right)$$ 19. $$\sin (195^{\circ})$$ 20. $$\sin (75^{\circ})$$ 21. $$\cos (345^{\circ})$$ 22. $$\cos (165^{\circ})$$ 23. $$\tan (-15^{\circ})$$ 24. $$\cos (105^{\circ})$$

$$\bigstar$$ Simplify

 25. $$\sin \left (x-\dfrac{3\pi }{4} \right)$$ 26. $$\sin \left (x+\dfrac{11\pi }{6} \right)$$ 27. $$\cos \left (x+\dfrac{2\pi }{3} \right)$$ 28. $$\cos \left (x-\dfrac{5\pi }{6} \right)$$
 29. $$\sec \left (\dfrac{\pi }{2}-\theta \right)$$ 30. $$\csc \left (\dfrac{\pi }{2}-t \right)$$ 31. $$\cot \left (\dfrac{\pi }{2}-x \right)$$ 32. $$\tan \left (\dfrac{\pi }{4}-x \right)$$

$$\bigstar$$ (a) Simplify and then (b) graph

 33. $$\cos \left ( \dfrac{\pi }{2}-x \right )$$ 34. $$\sin (\pi -x)$$ 35. $$\tan \left ( \dfrac{\pi }{3}+x \right )$$ 36. $$\sin \left ( \dfrac{\pi }{3}+x \right )$$ 37. $$\tan \left ( \dfrac{\pi }{4}-x \right )$$ 38. $$\cos \left ( \dfrac{7\pi }{6}+x \right )$$ 39. $$\sin \left ( \dfrac{\pi }{4}+x \right )$$ 40. $$\cos \left ( \dfrac{5\pi }{4}+x \right )$$

1.  $$\dfrac{\sqrt{2}+\sqrt{6}}{4}$$     3.  $$\dfrac{-\sqrt{2}-\sqrt{6}}{4}$$    5.  $$\dfrac{\sqrt{6}-\sqrt{2}}{4}$$    7.  $$\dfrac{\sqrt{2}-\sqrt{6}}{4}$$        9. $$-2 - \sqrt{3}$$     11. $$2 + \sqrt{3}$$

13. $$\dfrac{\sqrt{2}+\sqrt{6}}{4}$$     15. $$\dfrac{\sqrt{6}-\sqrt{2}}{4}$$     17. $$-2-\sqrt{3}$$  19. $$-\dfrac{\sqrt{3}-1}{2\sqrt{2}}$$     21. $$\dfrac{1+\sqrt{3}}{2\sqrt{2}}$$      23. $$-2 + \sqrt{3}$$

25. $$-\dfrac{\sqrt{2}}{2}\sin x-\dfrac{\sqrt{2}}{2}\cos x$$      27. $$-\dfrac{1}{2}\cos x-\dfrac{\sqrt{3}}{2}\sin x$$     29. $$\csc \theta$$     31. $$\tan x$$

33. $$\sin x$$

35. $$\cot \left ( \dfrac{\pi }{6}-x \right )$$

37. $$\cot \left ( \dfrac{\pi }{4}+x \right )$$

39. $$\dfrac{\sin x}{\sqrt{2}}+\dfrac{\cos x}{\sqrt{2}}$$

### B: Evaluate sum and difference formulas given trig ratios of angles

Exercise $$\PageIndex{B}$$

41. Given that $$\sin a=\dfrac{4}{5}$$ and $$\cos b=\dfrac{1}{3}$$, with $$a$$ and $$b$$ both in the interval $$\left [ 0, \dfrac{\pi }{2} \right )$$

(a) Find $$\sin (a-b)$$            (b) Find  $$\cos (a+b)$$.

42. Given that $$\sin a=\dfrac{2}{3}$$ and $$\cos b=-\dfrac{1}{4}$$$,$ with $$a$$ and $$b$$ both in the interval $$\left [ \dfrac{\pi }{2}, \pi \right )$$$,$

(a) Find $$\sin (a+b)$$           (b) Find $$\cos (a-b)$$.

43. Angles $$A$$ and $$B$$ are in standard position and $$\sin( A ) = \dfrac{1}{2}, \cos(A) > 0)$$, $$\cos(B) = \dfrac{3}{4}$$, and $$\sin(B) < 0$$.
Draw a picture of the angles $$A$$ and $$B$$ in the plane and then find each of the following.

(a) $$\cos(A + B)$$          (b) $$\cos(A - B)$$          (c) $$\sin(A + B)$$          (d) $$\sin(A - B)$$          (e) $$\tan(A + B)$$          (f) $$\tan(A - B)$$

$$\bigstar$$ Given the following information about angles $$A$$ and $$B$$, find the exact value for each of the following.

(a) $$\cos(A + B)$$          (b) $$\cos(A - B)$$          (c) $$\sin(A + B)$$          (d) $$\sin(A - B)$$          (e) $$\tan(A + B)$$          (f) $$\tan(A - B)$$

45. $$\sin A = -\dfrac{3}{5}$$ with $$A$$ in Quadrant III, and $$\cos B = \dfrac{1}{2}$$ with  $$B$$ in Quadrant IV.

46. $$\sin A = \dfrac{4}{5}$$ with $$A$$ in Quadrant I, and $$\tan B = -\dfrac{\sqrt{5}}{2}$$ with  $$B$$ in Quadrant II.

47. $$\cos A = \dfrac{1}{2}$$ with $$0 \le A \le \tfrac{\pi}{2}$$, and $$\tan B = 2 \sqrt{2}$$ with  $$\pi \le B \le \tfrac{3\pi}{2}$$.

48. $$\cos A = -\dfrac{\sqrt{3}}{3}$$ with $$\pi \le A \le \tfrac{3\pi}{2}$$, and $$\sin B = \dfrac{\sqrt{3}}{3}$$ with  $$\tfrac{\pi}{2} \le B \le \pi$$.

49. $$A = \tan^{-1} \left( \sqrt{5} \right)$$ and  $$B = \sin^{-1} \left( - \dfrac{\sqrt{5}}{3} \right)$$

50. $$A = \tan^{-1} \left( - 2 \right)$$ and $$B = \cos^{-1} \left( - \dfrac{\sqrt{6}}{6} \right)$$

$$\bigstar$$ Find the exact value of each expression.

 52. $$\quad \sin \left ( \cos^{-1}\left ( 0 \right )- \cos^{-1}\left ( \dfrac{1}{2} \right )\right )$$ 53. $$\quad \cos \left ( \cos^{-1}\left ( \dfrac{\sqrt{2}}{2} \right )+ \sin^{-1}\left ( \dfrac{\sqrt{3}}{2} \right )\right )$$ 54. $$\quad \tan \left ( \sin^{-1}\left ( \dfrac{1}{2} \right )- \cos^{-1}\left ( \dfrac{1}{2} \right )\right )$$

 58. $$\tan(\sin^{-1} u + \cos^{-1} u )$$ 59. $$\cos( \sin^{-1} u + \tan^{-1} u )$$ 60. $$\sin( \cos^{-1} u +\ tan^{-1} u )$$

41a. $$\sin (a-b)=\left ( \dfrac{4}{5} \right )\left ( \dfrac{1}{3} \right )-\left ( \dfrac{3}{5} \right )\left ( \dfrac{2\sqrt{2}}{3} \right )=\dfrac{4-6\sqrt{2}}{15}$$

41b. $$\cos (a+b)=\left ( \dfrac{3}{5} \right )\left ( \dfrac{1}{3} \right )-\left ( \dfrac{4}{5} \right )\left ( \dfrac{2\sqrt{2}}{3} \right )=\dfrac{3-8\sqrt{2}}{15}$$

43. Use the Pythagorean Identity to determine  $$\cos(A) = \dfrac{\sqrt{3}}{2}$$ and $$\sin(B) = -\dfrac{\sqrt{7}}{4}$$
43(a). $$\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) = \dfrac{\sqrt{3}}{2}\cdot\dfrac{3}{4} - \dfrac{1}{2}\cdot(-\dfrac{\sqrt{7}}{4}) = \dfrac{3\sqrt{3} + \sqrt{7}}{8}$$

45. $$\sin (A+B) = \tfrac{-3}{5} \cdot \tfrac{-4}{5} + \tfrac{-4}{5} \cdot \tfrac{\sqrt{3}}{2} = \dfrac{-3-4\sqrt{3}}{10}$$, $$\cos(A+B) = \dfrac{-4+3\sqrt{3}}{10}$$, $$\tan(A+B) = \dfrac{48 + 25\sqrt{3}}{-11} \\$$
$$\quad \;\; \sin(A-B) = \dfrac{-3+4\sqrt{3}}{10}$$, $$\cos(A-B) = \dfrac{-4-3\sqrt{3}}{10}$$, $$\tan(A-B) = \dfrac{48 + 25\sqrt{3}}{-11}$$

47. $$\sin (A+B) = \tfrac{\sqrt{3}}{2} \cdot \tfrac{-1}{3} + \tfrac{1}{2} \cdot \tfrac{-2\sqrt{2}}{3} = \dfrac{-\sqrt{3}-2\sqrt{2}}{6}$$, $$\cos(A+B) = \dfrac{-1+2\sqrt{6}}{6}$$, $$\tan(A+B) = \dfrac{-9\sqrt{3} - 8\sqrt{2}}{23} \\$$
$$\quad \;\; \sin(A-B) = \dfrac{-\sqrt{3}+2\sqrt{2}}{6}$$, $$\cos(A-B) = \dfrac{-1-2\sqrt{6}}{6}$$, $$\tan(A-B) = \dfrac{-9\sqrt{3} + 8\sqrt{2}}{23}$$

49. $$\sin (A+B) = \tfrac{\sqrt{3}}{6} \cdot \tfrac{2}{3} + \tfrac{\sqrt{6}}{6} \cdot \tfrac{-\sqrt{5}}{3} = \dfrac{\sqrt{30}}{18},$$ $$\cos(A+B) = \dfrac{7\sqrt{6}}{18},$$ $$\tan(A+B) = \dfrac{\sqrt{5}}{7} \\$$
$$\quad \;\; \sin(A-B) = \dfrac{\sqrt{30}}{6}$$, $$\cos(A-B) = \dfrac{-\sqrt{6}}{6}$$, $$\tan(A-B) = -\sqrt{5}$$

53. $$\dfrac{\sqrt{2}-\sqrt{6}}{4}$$       59. $$\dfrac{\sqrt{1-u^4}}{1+u^2} -u^2$$

### C: Solve Equations

Exercise $$\PageIndex{D}$$

$$\bigstar$$ Solve each equation for all solutions.

 65. $$\sin \left(3x\right)\cos \left(6x\right)-\cos \left(3x\right)\sin \left(6x\right)= -0.9$$ 66. $$\sin \left(6x\right)\cos \left(11x\right)-\cos \left(6x\right)\sin \left(11x\right)= -0.1$$ 67. $$\cos \left(2x\right)\cos \left(x\right)+\sin \left(2x\right)\sin \left(x\right)=1$$ 68. $$\cos \left(5x\right)\cos \left(3x\right)-\sin \left(5x\right)\sin \left(3x\right)=\dfrac{\sqrt{3} }{2}$$

65. $$0.373 + \dfrac{2\pi}{3} k$$ and $$0.674 + \dfrac{2\pi}{3} k$$, where $$k$$ is an integer     67. $$2 \pi k$$, where $$k$$ is an integer

### D: Verify Identities

Exercise $$\PageIndex{F}$$

$$\bigstar$$ Simplify.

 71.  $$\dfrac{\tan \left (\dfrac{3}{2}x \right)-\tan \left (\dfrac{7}{5}x \right)}{1+\tan \left (\dfrac{3}{2}x \right)\tan \left (\dfrac{7}{5}x \right)}$$ 72. $$\sin(2x)\cos(5x)-\sin(5x)\cos(2x)$$

$$\bigstar$$ Verify the Identity.

 73. $$\dfrac{\sin \left(x\right)+\sin \left(y\right)}{\cos \left(x\right)+\cos \left(y\right)} =\tan \left(\dfrac{1}{2} \left(x+y\right)\right)$$ 74. $$\dfrac{\sin\;(A-B)}{\sin\;(A+B)} ~=~ \dfrac{\cot\;B \;-\; \cot\;A}{\cot\;B \;+\; \cot\;A}$$ 75. $$\cot\;A ~+~ \cot\;B ~=~ \dfrac{\sin\;(A+B)}{\sin\;A~\sin\;B}$$ 76. $$\dfrac{\sin(r + s)}{\cos(r)\cos(s)} = \tan(r) + \tan(s)$$ 77. $$\dfrac{\sin(r - s)}{\cos(r)\cos(s)} = \tan(r) - \tan(s)$$ 78. $$\dfrac{\cos\;(A+B)}{\sin\;A~\cos\;B} ~=~ \cot\;A \;-\; \tan\;B$$ 79. $$\dfrac{\cos (a+b)}{\cos a \cos b}=1-\tan a \tan b$$ 80. $$\dfrac{\tan (a+b)}{\tan (a-b)}=\dfrac{\sin a \cos a + \sin b \cos b}{\sin a \cos a - \sin b \cos b}$$ 81. $$\tan \left ( x+\dfrac{\pi }{4} \right )=\dfrac{\tan x+1}{1-\tan x}$$ 82. $$\tan \left(\dfrac{\pi }{4} -t\right)=\dfrac{1-\tan \left(t\right)}{1+\tan \left(t\right)}$$ 83. $$\dfrac{\tan (x+y)}{1+\tan x \tan x}=\dfrac{\tan x + \tan y}{1-\tan^2 x \tan^2 y}$$ 84. $$\cot\;(A+B) ~=~ \dfrac{\cot\;A~\cot\;B \;-\; 1}{\cot\;A \;+\; \cot\;B}$$ 85. $$\cot\;(A-B) ~=~ \dfrac{\cot\;A~\cot\;B \;+\; 1}{\cot\;B \;-\; \cot\;A}$$ 87. $$\dfrac{\cos(x+h)-\cos(x)}{h}=\cos x\dfrac{\cos (h)-1}{h}-\sin x \dfrac{\sin (h)}{h}$$
71. $$\tan \left (\dfrac{x}{10} \right)$$

75. $$\cot\;A ~+~ \cot\;B = \text{___________} = \dfrac{\sin\;(A+B)}{\sin\;A~\sin\;B}$$
77. $$\dfrac{\sin(r - s)}{\cos(r)\cos(s)} = \text{___________} = \tan(r) - \tan(s)$$
79. $$\dfrac{\cos (a+b)}{\cos a \cos b} = \dfrac{\cos a \cos b}{\cos a \cos b}- \dfrac{\sin a \sin b}{\cos a \cos b}= 1-\tan a \tan b$$
81. $$\tan \left ( x+\dfrac{\pi }{4} \right ) = \dfrac{\tan x + \tan\left (\tfrac{\pi}{4} \right )}{1-\tan x \tan\left (\tfrac{\pi}{4} \right )} = \dfrac{\tan x+1}{1-\tan x(1)} = \dfrac{\tan x+1}{1-\tan x}$$
83. $$\dfrac{\tan (x+y)}{1+\tan x \tan x}= \text{___________} = \dfrac{\tan x + \tan y}{1-\tan^2 x \tan^2 y}$$
85. $$\cot\;(A-B) = \text{___________} = \dfrac{\cot\;A~\cot\;B \;+\; 1}{\cot\;B \;-\; \cot\;A}$$
87. $$\dfrac{\cos(x+h)-\cos(x)}{h} = \dfrac{\cos x\cos (h) - \sin x\sin (h) -\cos x}{h} = \dfrac{\cos x(\cos (h)-1) - \sin x(\sin (h))}{h} \\$$
$$\quad \;\; = \cos x \cdot \dfrac{\cos (h)-1}{h}-\sin x \cdot \dfrac{\sin (h)}{h}$$

$$\bigstar$$ Verify the Identity.

 90. $$2\sin \left(a+b\right)\sin \left(a-b\right)=\cos \left(2b\right)-{\rm cos}(2a)$$ 91. $$\cos(x+y)\cos(x-y)=\cos^2x-\sin^2y$$ 92. $$\cos \left(a+b\right)+\cos \left(a-b\right)=2\cos \left(a\right)\cos \left(b\right)$$ 93. $$\sin(4x)-\sin(3x)\cos x =\sin x \cos(3x)$$ 94. $$\cos(4x)+\sin x \sin(3x) = \cos x \cos(3x)$$ 95. $$\sin(3x)\cos(6x)=\sin(9x)-\cos(3x)\sin(6x)$$  96. $$\sin(4x) = \sin(5x)\cos x-\cos(5x)\sin x$$ 97. $$\sin(3x)=3\sin x \cos^2x-\sin^3x$$ 98. $$\cos(3x)=\cos^3x-3\sin^2x\cos x$$ 99. $$\sin(2x) = 2 \sin x \cos x$$ 100. $$\cos(2\theta ) = \cos^2\theta -\sin^2\theta$$ 101. $$\tan(2\theta ) = \dfrac{2\tan \theta }{1-\tan^2\theta }$$ 102. $$\tan(-x)=\dfrac{\tan x-\tan(2x)}{1+\tan x \tan(2x)}$$
91. $$\cos(x+y)\cos(x-y) = \text{___________} = \cos^2x-\sin^2y$$
93. $$\sin(4x)-\sin(3x)\cos x = \text{___________} = \sin x \cos(3x)$$
95. $$\sin(3x)\cos(6x) = \text{___________} = \sin(9x)-\cos(3x)\sin(6x)$$
97. $$\sin (x+2x) = \sin x \cos (2x)+\sin (2x) \cos x = \sin x(\cos ^2 x - \sin ^2 x)+2\sin x \cos x \cos x \\ = \sin x \cos ^2 x-\sin ^3 x + 2\sin x\cos ^2 x = 3\sin x\cos ^2 x - \sin ^3 x$$
99. $$\sin(2x) = \text{___________} = 2 \sin x \cos x$$
101. $$\tan(2\theta ) = \text{___________} = \dfrac{2\tan \theta }{1-\tan^2\theta }$$
$$\bigstar$$