1. Explain how to determine the reduction identities from the double-angle identity [latex]\cos \left(2x\right)={\cos }^{2}x-{\sin }^{2}x[/latex].

2. Explain how to determine the double-angle formula for [latex]\tan \left(2x\right)[/latex] using the double-angle formulas for [latex]\cos \left(2x\right)[/latex] and [latex]\sin \left(2x\right)[/latex].

3. We can determine the half-angle formula for [latex]\tan \left(\frac{x}{2}\right)=\frac{\sqrt{1-\cos x}}{\sqrt{1+\cos x}}[/latex] by dividing the formula for [latex]\sin \left(\frac{x}{2}\right)[/latex] by [latex]\cos \left(\frac{x}{2}\right)[/latex]. Explain how to determine two formulas for [latex]\tan \left(\frac{x}{2}\right)[/latex] that do not involve any square roots.

4. For the half-angle formula given in the previous exercise for [latex]\tan \left(\frac{x}{2}\right)[/latex], explain why dividing by 0 is not a concern. (Hint: examine the values of [latex]\cos x[/latex] necessary for the denominator to be 0.)

For the following exercises, find the exact values of a) [latex]\sin \left(2x\right)[/latex], b) [latex]\cos \left(2x\right)[/latex], and c) [latex]\tan \left(2x\right)[/latex] without solving for [latex]x[/latex].

5. If [latex]\sin x=\frac{1}{8}[/latex], and [latex]x[/latex] is in quadrant I.

6. If [latex]\cos x=\frac{2}{3}[/latex], and [latex]x[/latex] is in quadrant I.

7. If [latex]\cos x=-\frac{1}{2}[/latex], and [latex]x[/latex] is in quadrant III.

8. If [latex]\tan x=-8[/latex], and [latex]x[/latex] is in quadrant IV.

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.

9. [latex]\cos \left(2\theta \right)=\frac{3}{5}[/latex] and [latex]{90}^{\circ }\le \theta \le {180}^{\circ }[/latex]

10. [latex]\cos \left(2\theta \right)=\frac{1}{\sqrt{2}}[/latex] and [latex]{180}^{\circ }\le \theta \le {270}^{\circ }[/latex]

For the following exercises, simplify to one trigonometric expression.

11. [latex]2\sin \left(\frac{\pi }{4}\right)2\cos \left(\frac{\pi }{4}\right)[/latex]

12. [latex]4\sin \left(\frac{\pi }{8}\right)\cos \left(\frac{\pi }{8}\right)[/latex]

For the following exercises, find the exact value using half-angle formulas.

13. [latex]\sin \left(\frac{\pi }{8}\right)[/latex]

14. [latex]\cos \left(-\frac{11\pi }{12}\right)[/latex]

15. [latex]\sin \left(\frac{11\pi }{12}\right)[/latex]

16. [latex]\cos \left(\frac{7\pi }{8}\right)[/latex]

17. [latex]\tan \left(\frac{5\pi }{12}\right)[/latex]

18. [latex]\tan \left(-\frac{3\pi }{12}\right)[/latex]

19. [latex]\tan \left(-\frac{3\pi }{8}\right)[/latex]

For the following exercises, find the exact values of a) [latex]\sin \left(\frac{x}{2}\right)[/latex], b) [latex]\cos \left(\frac{x}{2}\right)[/latex], and c) [latex]\tan \left(\frac{x}{2}\right)[/latex] without solving for [latex]x[/latex].

20. If [latex]\tan x=-\frac{4}{3}[/latex], and [latex]x[/latex] is in quadrant IV.

21. If [latex]\sin x=-\frac{12}{13}[/latex], and [latex]x[/latex] is in quadrant III.

22. If [latex]\csc x=7[/latex], and [latex]x[/latex] is in quadrant II.

23. If [latex]\sec x=-4[/latex], and [latex]x[/latex] is in quadrant II.

For the following exercises, use Figure 5 to find the requested half and double angles.

24. Find [latex]\sin \left(2\theta \right),\cos \left(2\theta \right)[/latex], and [latex]\tan \left(2\theta \right)[/latex].

25. Find [latex]\sin \left(2\alpha \right),\cos \left(2\alpha \right)[/latex], and [latex]\tan \left(2\alpha \right)[/latex].

26. Find [latex]\sin \left(\frac{\theta }{2}\right),\cos \left(\frac{\theta }{2}\right)[/latex], and [latex]\tan \left(\frac{\theta }{2}\right)[/latex].

27. Find [latex]\sin \left(\frac{\alpha }{2}\right),\cos \left(\frac{\alpha }{2}\right)[/latex], and [latex]\tan \left(\frac{\alpha }{2}\right)[/latex].

For the following exercises, simplify each expression. Do not evaluate.

28. [latex]{\cos }^{2}\left({28}^{\circ }\right)-{\sin }^{2}\left({28}^{\circ }\right)[/latex]

29. [latex]2{\cos }^{2}\left({37}^{\circ }\right)-1[/latex]

30. [latex]1 - 2{\sin }^{2}\left({17}^{\circ }\right)[/latex]

31. [latex]{\cos }^{2}\left(9x\right)-{\sin }^{2}\left(9x\right)[/latex]

32. [latex]4\sin \left(8x\right)\cos \left(8x\right)[/latex]

33. [latex]6\sin \left(5x\right)\cos \left(5x\right)[/latex]

For the following exercises, prove the identity given.

34. [latex]{\left(\sin t-\cos t\right)}^{2}=1-\sin \left(2t\right)[/latex]

35. [latex]\sin \left(2x\right)=-2\sin \left(-x\right)\cos \left(-x\right)[/latex]

36. [latex]\cot x-\tan x=2\cot \left(2x\right)[/latex]

37. [latex]\frac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)}{\tan }^{2}\theta =\tan \theta[/latex]

For the following exercises, rewrite the expression with an exponent no higher than 1.

38. [latex]{\cos }^{2}\left(5x\right)[/latex]

39. [latex]{\cos }^{2}\left(6x\right)[/latex]

40. [latex]{\sin }^{4}\left(8x\right)[/latex]

41. [latex]{\sin }^{4}\left(3x\right)[/latex]

42. [latex]{\cos }^{2}x{\sin }^{4}x[/latex]

43. [latex]{\cos }^{4}x{\sin }^{2}x[/latex]

44. [latex]{\tan }^{2}x{\sin }^{2}x[/latex]

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.

45. [latex]{\tan }^{4}x[/latex]

46. [latex]{\sin }^{2}\left(2x\right)[/latex]

47. [latex]{\sin }^{2}x{\cos }^{2}x[/latex]

48. [latex]{\tan }^{2}x\sin x[/latex]

49. [latex]{\tan }^{4}x{\cos }^{2}x[/latex]

50. [latex]{\cos }^{2}x\sin \left(2x\right)[/latex]

51. [latex]{\cos }^{2}\left(2x\right)\sin x[/latex]

52. [latex]{\tan }^{2}\left(\frac{x}{2}\right)\sin x[/latex]

For the following exercises, algebraically find an equivalent function, only in terms of [latex]\sin x[/latex] and/or [latex]\cos x[/latex], and then check the answer by graphing both equations.

53. [latex]\sin \left(4x\right)[/latex]

54. [latex]\cos \left(4x\right)[/latex]

For the following exercises, prove the identities.

55. [latex]\sin \left(2x\right)=\frac{2\tan x}{1+{\tan }^{2}x}[/latex]

56. [latex]\cos \left(2\alpha \right)=\frac{1-{\tan }^{2}\alpha }{1+{\tan }^{2}\alpha }[/latex]

57. [latex]\tan \left(2x\right)=\frac{2\sin x\cos x}{2{\cos }^{2}x - 1}[/latex]

58. [latex]{\left({\sin }^{2}x - 1\right)}^{2}=\cos \left(2x\right)+{\sin }^{4}x[/latex]

59. [latex]\sin \left(3x\right)=3\sin x{\cos }^{2}x-{\sin }^{3}x[/latex]

60. [latex]\cos \left(3x\right)={\cos }^{3}x - 3{\sin }^{2}x\cos x[/latex]

61. [latex]\frac{1+\cos \left(2t\right)}{\sin \left(2t\right)-\cos t}=\frac{2\cos t}{2\sin t - 1}[/latex]

62. [latex]\sin \left(16x\right)=16\sin x\cos x\cos \left(2x\right)\cos \left(4x\right)\cos \left(8x\right)[/latex]

63. [latex]\cos \left(16x\right)=\left({\cos }^{2}\left(4x\right)-{\sin }^{2}\left(4x\right)-\sin \left(8x\right)\right)\left({\cos }^{2}\left(4x\right)-{\sin }^{2}\left(4x\right)+\sin \left(8x\right)\right)[/latex]