## Problem Set 54: Sum-to-Product and Product-to-Sum Formulas

1. Starting with the product to sum formula $\sin \alpha \cos \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right]$, explain how to determine the formula for $\cos \alpha \sin \beta$.

2. Explain two different methods of calculating $\cos \left(195^\circ \right)\cos \left(105^\circ \right)$, one of which uses the product to sum. Which method is easier?

3. Explain a situation where we would convert an equation from a sum to a product and give an example.

4. Explain a situation where we would convert an equation from a product to a sum, and give an example.

For the following exercises, rewrite the product as a sum or difference.

5. $16\sin \left(16x\right)\sin \left(11x\right)$

6. $20\cos \left(36t\right)\cos \left(6t\right)$

7. $2\sin \left(5x\right)\cos \left(3x\right)$

8. $10\cos \left(5x\right)\sin \left(10x\right)$

9. $\sin \left(-x\right)\sin \left(5x\right)$

10. $\sin \left(3x\right)\cos \left(5x\right)$

For the following exercises, rewrite the sum or difference as a product.

11. $\cos \left(6t\right)+\cos \left(4t\right)$

12. $\sin \left(3x\right)+\sin \left(7x\right)$

13. $\cos \left(7x\right)+\cos \left(-7x\right)$

14. $\sin \left(3x\right)-\sin \left(-3x\right)$

15. $\cos \left(3x\right)+\cos \left(9x\right)$

16. $\sin h-\sin \left(3h\right)$

For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

17. $\cos \left(45^\circ \right)\cos \left(15^\circ \right)$

18. $\cos \left(45^\circ \right)\sin \left(15^\circ \right)$

19. $\sin \left(-345^\circ \right)\sin \left(-15^\circ \right)$

20. $\sin \left(195^\circ \right)\cos \left(15^\circ \right)$

21. $\sin \left(-45^\circ \right)\sin \left(-15^\circ \right)$

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

22. $\cos \left(23^\circ \right)\sin \left(17^\circ \right)$

23. $2\sin \left(100^\circ \right)\sin \left(20^\circ \right)$

24. $2\sin \left(-100^\circ \right)\sin \left(-20^\circ \right)$

25. $\sin \left(213^\circ \right)\cos \left(8^\circ \right)$

26. $2\cos \left(56^\circ \right)\cos \left(47^\circ \right)$

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

27. $\sin \left(76^\circ \right)+\sin \left(14^\circ \right)$

28. $\cos \left(58^\circ \right)-\cos \left(12^\circ \right)$

29. $\sin \left(101^\circ \right)-\sin \left(32^\circ \right)$

30. $\cos \left(100^\circ \right)+\cos \left(200^\circ \right)$

31. $\sin \left(-1^\circ \right)+\sin \left(-2^\circ \right)$

For the following exercises, prove the identity.

32. $\frac{\cos \left(a+b\right)}{\cos \left(a-b\right)}=\frac{1-\tan a\tan b}{1+\tan a\tan b}$

33. $4\sin \left(3x\right)\cos \left(4x\right)=2\sin \left(7x\right)-2\sin x$

34. $\frac{6\cos \left(8x\right)\sin \left(2x\right)}{\sin \left(-6x\right)}=-3\sin \left(10x\right)\csc \left(6x\right)+3$

35. $\sin x+\sin \left(3x\right)=4\sin x{\cos }^{2}x$

36. $2\left({\cos }^{3}x-\cos x{\sin }^{2}x\right)=\cos \left(3x\right)+\cos x$

37. $2\tan x\cos \left(3x\right)=\sec x\left(\sin \left(4x\right)-\sin \left(2x\right)\right)$

38. $\cos \left(a+b\right)+\cos \left(a-b\right)=2\cos a\cos b$

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

39. $\cos \left({58}^{\circ }\right)+\cos \left({12}^{\circ }\right)$

40. $\sin \left({2}^{\circ }\right)-\sin \left({3}^{\circ }\right)$

41. $\cos \left({44}^{\circ }\right)-\cos \left({22}^{\circ }\right)$

42. $\cos \left({176}^{\circ }\right)\sin \left({9}^{\circ }\right)$

43. $\sin \left(-{14}^{\circ }\right)\sin \left({85}^{\circ }\right)$

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.

44. $2\sin \left(2x\right)\sin \left(3x\right)=\cos x-\cos \left(5x\right)$

45. $\frac{\cos \left(10\theta \right)+\cos \left(6\theta \right)}{\cos \left(6\theta \right)-\cos \left(10\theta \right)}=\cot \left(2\theta \right)\cot \left(8\theta \right)$

46. $\frac{\sin \left(3x\right)-\sin \left(5x\right)}{\cos \left(3x\right)+\cos \left(5x\right)}=\tan x$

47. $2\cos \left(2x\right)\cos x+\sin \left(2x\right)\sin x=2\sin x$

48. $\frac{\sin \left(2x\right)+\sin \left(4x\right)}{\sin \left(2x\right)-\sin \left(4x\right)}=-\tan \left(3x\right)\cot x$

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

49. $\frac{\sin \left(9t\right)-\sin \left(3t\right)}{\cos \left(9t\right)+\cos \left(3t\right)}$

50. $2\sin \left(8x\right)\cos \left(6x\right)-\sin \left(2x\right)$

51. $\frac{\sin \left(3x\right)-\sin x}{\sin x}$

52. $\frac{\cos \left(5x\right)+\cos \left(3x\right)}{\sin \left(5x\right)+\sin \left(3x\right)}$

53. $\sin x\cos \left(15x\right)-\cos x\sin \left(15x\right)$

For the following exercises, prove the following sum-to-product formulas.

54. $\sin x-\sin y=2\sin \left(\frac{x-y}{2}\right)\cos \left(\frac{x+y}{2}\right)$

55. $\cos x+\cos y=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

For the following exercises, prove the identity.

56. $\frac{\sin \left(6x\right)+\sin \left(4x\right)}{\sin \left(6x\right)-\sin \left(4x\right)}=\tan \left(5x\right)\cot x$

57. $\frac{\cos \left(3x\right)+\cos x}{\cos \left(3x\right)-\cos x}=-\cot \left(2x\right)\cot x$

58. $\frac{\cos \left(6y\right)+\cos \left(8y\right)}{\sin \left(6y\right)-\sin \left(4y\right)}=\cot y\cos \left(7y\right)\sec \left(5y\right)$

59. $\frac{\cos \left(2y\right)-\cos \left(4y\right)}{\sin \left(2y\right)+\sin \left(4y\right)}=\tan y$

60. $\frac{\sin \left(10x\right)-\sin \left(2x\right)}{\cos \left(10x\right)+\cos \left(2x\right)}=\tan \left(4x\right)$

61. $\cos x-\cos \left(3x\right)=4{\sin }^{2}x\cos x$

62. ${\left(\cos \left(2x\right)-\cos \left(4x\right)\right)}^{2}+{\left(\sin \left(4x\right)+\sin \left(2x\right)\right)}^{2}=4{\sin }^{2}\left(3x\right)$

63. $\tan \left(\frac{\pi }{4}-t\right)=\frac{1-\tan t}{1+\tan t}$