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

2. Explain two different methods of calculating [latex]\cos \left(195^\circ \right)\cos \left(105^\circ \right)[/latex], 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. [latex]16\sin \left(16x\right)\sin \left(11x\right)[/latex]

6. [latex]20\cos \left(36t\right)\cos \left(6t\right)[/latex]

7. [latex]2\sin \left(5x\right)\cos \left(3x\right)[/latex]

8. [latex]10\cos \left(5x\right)\sin \left(10x\right)[/latex]

9. [latex]\sin \left(-x\right)\sin \left(5x\right)[/latex]

10. [latex]\sin \left(3x\right)\cos \left(5x\right)[/latex]

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

11. [latex]\cos \left(6t\right)+\cos \left(4t\right)[/latex]

12. [latex]\sin \left(3x\right)+\sin \left(7x\right)[/latex]

13. [latex]\cos \left(7x\right)+\cos \left(-7x\right)[/latex]

14. [latex]\sin \left(3x\right)-\sin \left(-3x\right)[/latex]

15. [latex]\cos \left(3x\right)+\cos \left(9x\right)[/latex]

16. [latex]\sin h-\sin \left(3h\right)[/latex]

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

17. [latex]\cos \left(45^\circ \right)\cos \left(15^\circ \right)[/latex]

18. [latex]\cos \left(45^\circ \right)\sin \left(15^\circ \right)[/latex]

19. [latex]\sin \left(-345^\circ \right)\sin \left(-15^\circ \right)[/latex]

20. [latex]\sin \left(195^\circ \right)\cos \left(15^\circ \right)[/latex]

21. [latex]\sin \left(-45^\circ \right)\sin \left(-15^\circ \right)[/latex]

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

22. [latex]\cos \left(23^\circ \right)\sin \left(17^\circ \right)[/latex]

23. [latex]2\sin \left(100^\circ \right)\sin \left(20^\circ \right)[/latex]

24. [latex]2\sin \left(-100^\circ \right)\sin \left(-20^\circ \right)[/latex]

25. [latex]\sin \left(213^\circ \right)\cos \left(8^\circ \right)[/latex]

26. [latex]2\cos \left(56^\circ \right)\cos \left(47^\circ \right)[/latex]

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

27. [latex]\sin \left(76^\circ \right)+\sin \left(14^\circ \right)[/latex]

28. [latex]\cos \left(58^\circ \right)-\cos \left(12^\circ \right)[/latex]

29. [latex]\sin \left(101^\circ \right)-\sin \left(32^\circ \right)[/latex]

30. [latex]\cos \left(100^\circ \right)+\cos \left(200^\circ \right)[/latex]

31. [latex]\sin \left(-1^\circ \right)+\sin \left(-2^\circ \right)[/latex]

For the following exercises, prove the identity.

32. [latex]\frac{\cos \left(a+b\right)}{\cos \left(a-b\right)}=\frac{1-\tan a\tan b}{1+\tan a\tan b}[/latex]

33. [latex]4\sin \left(3x\right)\cos \left(4x\right)=2\sin \left(7x\right)-2\sin x[/latex]

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

35. [latex]\sin x+\sin \left(3x\right)=4\sin x{\cos }^{2}x[/latex]

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

37. [latex]2\tan x\cos \left(3x\right)=\sec x\left(\sin \left(4x\right)-\sin \left(2x\right)\right)[/latex]

38. [latex]\cos \left(a+b\right)+\cos \left(a-b\right)=2\cos a\cos b[/latex]

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. [latex]\cos \left({58}^{\circ }\right)+\cos \left({12}^{\circ }\right)[/latex]

40. [latex]\sin \left({2}^{\circ }\right)-\sin \left({3}^{\circ }\right)[/latex]

41. [latex]\cos \left({44}^{\circ }\right)-\cos \left({22}^{\circ }\right)[/latex]

42. [latex]\cos \left({176}^{\circ }\right)\sin \left({9}^{\circ }\right)[/latex]

43. [latex]\sin \left(-{14}^{\circ }\right)\sin \left({85}^{\circ }\right)[/latex]

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. [latex]2\sin \left(2x\right)\sin \left(3x\right)=\cos x-\cos \left(5x\right)[/latex]

45. [latex]\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)[/latex]

46. [latex]\frac{\sin \left(3x\right)-\sin \left(5x\right)}{\cos \left(3x\right)+\cos \left(5x\right)}=\tan x[/latex]

47. [latex]2\cos \left(2x\right)\cos x+\sin \left(2x\right)\sin x=2\sin x[/latex]

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

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. [latex]\frac{\sin \left(9t\right)-\sin \left(3t\right)}{\cos \left(9t\right)+\cos \left(3t\right)}[/latex]

50. [latex]2\sin \left(8x\right)\cos \left(6x\right)-\sin \left(2x\right)[/latex]

51. [latex]\frac{\sin \left(3x\right)-\sin x}{\sin x}[/latex]

52. [latex]\frac{\cos \left(5x\right)+\cos \left(3x\right)}{\sin \left(5x\right)+\sin \left(3x\right)}[/latex]

53. [latex]\sin x\cos \left(15x\right)-\cos x\sin \left(15x\right)[/latex]

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

54. [latex]\sin x-\sin y=2\sin \left(\frac{x-y}{2}\right)\cos \left(\frac{x+y}{2}\right)[/latex]

55. [latex]\cos x+\cos y=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)[/latex]

For the following exercises, prove the identity.

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

57. [latex]\frac{\cos \left(3x\right)+\cos x}{\cos \left(3x\right)-\cos x}=-\cot \left(2x\right)\cot x[/latex]

58. [latex]\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)[/latex]

59. [latex]\frac{\cos \left(2y\right)-\cos \left(4y\right)}{\sin \left(2y\right)+\sin \left(4y\right)}=\tan y[/latex]

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

61. [latex]\cos x-\cos \left(3x\right)=4{\sin }^{2}x\cos x[/latex]

62. [latex]{\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)[/latex]

63. [latex]\tan \left(\frac{\pi }{4}-t\right)=\frac{1-\tan t}{1+\tan t}[/latex]