SECTION 7.4 Trigonometric Identities 509 In Problems 21–100, establish each identity. 21. csc cos cot θ θ θ ⋅ = 22. sec sin tan θ θ θ ⋅ = 23. 1 tan sec 2 2 θ θ ( ) + − = 24. 1 cot csc 2 2 θ θ ( ) + − = 25. cos tan cot csc θ θ θ θ ( ) + = 26. sin cot tan sec θ θ θ θ ( ) + = 27. u u u u tan cot cos sin 2 2 − = 28. u u u u sin csc cos sin 2 2 − = 29. sec 1 sec 1 tan2 θ θ θ ( )( ) − + = 30. csc 1 csc 1 cot2 θ θ θ ( )( ) − + = 31. sec tan sec tan 1 θ θ θ θ ( )( ) + − = 32. csc cot csc cot 1 θ θ θ θ ( )( ) + − = 33. cos 1 tan 1 2 2 θ θ ( ) + = 34. 1 cos 1 cot 1 2 2 θ θ ( )( ) − + = 35. sin cos sin cos 2 2 2 θ θ θ θ ( ) ( ) + + − = 36. tan cos cot sin 1 2 2 2 2 θ θ θ θ + = 37. sec sec tan tan 4 2 4 2 θ θ θ θ − = + 38. csc csc cot cot 4 2 4 2 θ θ θ θ − = + 39. u u u u sec tan cos 1 sin − = + 40. u u u u csc cot sin 1 cos − = + 41. 3sin 4cos 3 cos 2 2 2 θ θ θ + = + 42. 9 sec 5 tan 5 4 sec 2 2 2 θ θ θ − = + 43. 1 cos 1 sin sin 2 θ θ θ − + = 44. 1 sin 1 cos cos 2 θ θ θ − − = − 45. v v v v 1 tan 1 tan cot 1 cot 1 + − = + − 46. v v v v csc 1 csc 1 1 sin 1 sin − + = − + 47. sec csc sin cos 2 tan θ θ θ θ θ + = 48. csc 1 cot cot csc 1 θ θ θ θ − = + 49. 1 sin 1 sin csc 1 csc 1 θ θ θ θ + − = + − 50. cos 1 cos 1 1 sec 1 sec θ θ θ θ + − = + − 51. v v v v v 1 sin cos cos 1 sin 2 sec − + − = 52. v v v v v cos 1 sin 1 sin cos 2 sec + + + = 53. sin sin cos 1 1 cot θ θ θ θ − = − 54. 1 sin 1 cos cos 2 θ θ θ − + = 55. 1 sin 1 sin sec tan 2 θ θ θ θ ( ) − + = − 56. 1 cos 1 cos csc cot 2 θ θ θ θ ( ) − + = − 57. cos 1 tan sin 1 cot sin cos θ θ θ θ θ θ − + − = + 58. cot 1 tan tan 1 cot 1 tan cot θ θ θ θ θ θ − + − = + + 59. tan cos 1 sin sec θ θ θ θ + + = 60. sin cos cos sin tan 1 tan 2 2 2 θ θ θ θ θ θ − = − 61. tan sec 1 tan sec 1 tan sec θ θ θ θ θ θ + − − + = + 62. sin cos 1 sin cos 1 sin 1 cos θ θ θ θ θ θ − + + − = + 63. tan cot tan cot sin cos 2 2 θ θ θ θ θ θ − + = − 64. sec cos sec cos sin 1 cos 2 2 θ θ θ θ θ θ − + = + 65. u u u u u tan cot tan cot 1 2 sin2 − + + = 66. u u u u u tan cot tan cot 2cos 1 2 − + + = 67. sec tan cot cos tan sec θ θ θ θ θ θ + + = 68. sec 1 sec 1 cos sin2 θ θ θ θ + = − 69. 1 tan 1 tan 1 2 cos 2 2 2 θ θ θ − + + = 70. 1 cot 1 cot 2cos 1 2 2 2 θ θ θ − + + = 71. sec csc sec csc sin cos θ θ θ θ θ θ − = − 72. sin tan cos cot tan 2 2 2 θ θ θ θ θ − − = 73. sec cos sin tan θ θ θ θ − = 74. tan cot sec csc θ θ θ θ + = 75. 1 1 sin 1 1 sin 2 sec2 θ θ θ − + + = 76. 1 sin 1 sin 1 sin 1 sin 4tan sec θ θ θ θ θ θ + − − − + = Skill Building In Problems 11–20, simplify each trigonometric expression by following the indicated direction. 11. Rewrite in terms of sine and cosine functions: tan csc . θ θ ⋅ 12. Rewrite in terms of sine and cosine functions: cot sec . θ θ ⋅ 13. Multiply cos 1 sin θ θ − by 1 sin 1 sin . θ θ + + 14. Multiply sin 1 cos θ θ + by 1 cos 1 cos . θ θ − − 15. Rewrite as a single quotient: sin cos cos cos sin sin θ θ θ θ θ θ + + − 16. Rewrite as a single quotient: v v 1 1 cos 1 1 cos − + + 17. Multiply and simplify: sin cos sin cos 1 sin cos θ θ θ θ θ θ ( )( ) + + − 18. Multiply and simplify: tan 1 tan 1 sec tan 2 θ θ θ θ ( )( ) + + − 19. Factor and simplify: 3sin 4 sin 1 sin 2 sin 1 2 2 θ θ θ θ + + + + 20. Factor and simplify: cos 1 cos cos 2 2 θ θ θ − −
RkJQdWJsaXNoZXIy NjM5ODQ=