## Problem Set 53: Double Angle, Half Angle, and Reduction Formulas

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

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

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

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

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

5. If $\sin x=\frac{1}{8}$, and $x$ is in quadrant I.

6. If $\cos x=\frac{2}{3}$, and $x$ is in quadrant I.

7. If $\cos x=-\frac{1}{2}$, and $x$ is in quadrant III.

8. If $\tan x=-8$, and $x$ is in quadrant IV.

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

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

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

For the following exercises, simplify to one trigonometric expression.

11. $2\sin \left(\frac{\pi }{4}\right)2\cos \left(\frac{\pi }{4}\right)$

12. $4\sin \left(\frac{\pi }{8}\right)\cos \left(\frac{\pi }{8}\right)$

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

13. $\sin \left(\frac{\pi }{8}\right)$

14. $\cos \left(-\frac{11\pi }{12}\right)$

15. $\sin \left(\frac{11\pi }{12}\right)$

16. $\cos \left(\frac{7\pi }{8}\right)$

17. $\tan \left(\frac{5\pi }{12}\right)$

18. $\tan \left(-\frac{3\pi }{12}\right)$

19. $\tan \left(-\frac{3\pi }{8}\right)$

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

20. If $\tan x=-\frac{4}{3}$, and $x$ is in quadrant IV.

21. If $\sin x=-\frac{12}{13}$, and $x$ is in quadrant III.

22. If $\csc x=7$, and $x$ is in quadrant II.

23. If $\sec x=-4$, and $x$ is in quadrant II.

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

Figure 5

24. Find $\sin \left(2\theta \right),\cos \left(2\theta \right)$, and $\tan \left(2\theta \right)$.

25. Find $\sin \left(2\alpha \right),\cos \left(2\alpha \right)$, and $\tan \left(2\alpha \right)$.

26. Find $\sin \left(\frac{\theta }{2}\right),\cos \left(\frac{\theta }{2}\right)$, and $\tan \left(\frac{\theta }{2}\right)$.

27. Find $\sin \left(\frac{\alpha }{2}\right),\cos \left(\frac{\alpha }{2}\right)$, and $\tan \left(\frac{\alpha }{2}\right)$.

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

28. ${\cos }^{2}\left({28}^{\circ }\right)-{\sin }^{2}\left({28}^{\circ }\right)$

29. $2{\cos }^{2}\left({37}^{\circ }\right)-1$

30. $1 - 2{\sin }^{2}\left({17}^{\circ }\right)$

31. ${\cos }^{2}\left(9x\right)-{\sin }^{2}\left(9x\right)$

32. $4\sin \left(8x\right)\cos \left(8x\right)$

33. $6\sin \left(5x\right)\cos \left(5x\right)$

For the following exercises, prove the identity given.

34. ${\left(\sin t-\cos t\right)}^{2}=1-\sin \left(2t\right)$

35. $\sin \left(2x\right)=-2\sin \left(-x\right)\cos \left(-x\right)$

36. $\cot x-\tan x=2\cot \left(2x\right)$

37. $\frac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)}{\tan }^{2}\theta =\tan \theta$

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

38. ${\cos }^{2}\left(5x\right)$

39. ${\cos }^{2}\left(6x\right)$

40. ${\sin }^{4}\left(8x\right)$

41. ${\sin }^{4}\left(3x\right)$

42. ${\cos }^{2}x{\sin }^{4}x$

43. ${\cos }^{4}x{\sin }^{2}x$

44. ${\tan }^{2}x{\sin }^{2}x$

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

45. ${\tan }^{4}x$

46. ${\sin }^{2}\left(2x\right)$

47. ${\sin }^{2}x{\cos }^{2}x$

48. ${\tan }^{2}x\sin x$

49. ${\tan }^{4}x{\cos }^{2}x$

50. ${\cos }^{2}x\sin \left(2x\right)$

51. ${\cos }^{2}\left(2x\right)\sin x$

52. ${\tan }^{2}\left(\frac{x}{2}\right)\sin x$

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

53. $\sin \left(4x\right)$

54. $\cos \left(4x\right)$

For the following exercises, prove the identities.

55. $\sin \left(2x\right)=\frac{2\tan x}{1+{\tan }^{2}x}$

56. $\cos \left(2\alpha \right)=\frac{1-{\tan }^{2}\alpha }{1+{\tan }^{2}\alpha }$

57. $\tan \left(2x\right)=\frac{2\sin x\cos x}{2{\cos }^{2}x - 1}$

58. ${\left({\sin }^{2}x - 1\right)}^{2}=\cos \left(2x\right)+{\sin }^{4}x$

59. $\sin \left(3x\right)=3\sin x{\cos }^{2}x-{\sin }^{3}x$

60. $\cos \left(3x\right)={\cos }^{3}x - 3{\sin }^{2}x\cos x$

61. $\frac{1+\cos \left(2t\right)}{\sin \left(2t\right)-\cos t}=\frac{2\cos t}{2\sin t - 1}$

62. $\sin \left(16x\right)=16\sin x\cos x\cos \left(2x\right)\cos \left(4x\right)\cos \left(8x\right)$

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