694 CHAPTER 7 Trigonometric Identities and Equations 7.2 Exercises To the student: Exercises 1–44 are designed for practice in using the fundamental identities and applying algebraic techniques to trigonometric expressions. These skills are essential in verifying the identities that follow. CONCEPT PREVIEW Match each expression in Column I with its correct factorization in Column II. I 1. x2 - y2 2. x3 - y3 3. x3 + y3 4. x2 + 2xy + y2 II A. 1x + y21x2 - xy + y22 B. 1x + y21x - y2 C. 1x + y22 D. 1x - y21x2 + xy + y22 CONCEPT PREVIEWFill in the blank(s) to correctly complete each fundamental identity. 5. sin2 u + cos2 u = 6. tan2 u + 1 = 7. sin1-u2 = 8. sec1-u2 = 9. tanu = 1 = sinu 10. cot u = 1 = cos u Perform each indicated operation and simplify the result so that there are no quotients. 11. cot u + 1 cot u 12. sec x csc x + csc x sec x 13. tanx1cot x + csc x2 14. cos b1sec b + csc b2 15. 1 csc2 u + 1 sec2 u 16. cos x sec x + sinx csc x 17. 1sina - cos a22 18. 1tanx + cot x22 19. 11 + sint22 + cos2 t 20. 11 + tanu22 - 2 tanu 21. 1 1 + cos x - 1 1 - cos x 22. 1 sina - 1 - 1 sina + 1 Factor each trigonometric expression. 23. sin2 u - 1 24. sec2 u - 1 25. 1sinx + 122 - 1sinx - 122 26. 1tanx + cot x22 - 1tanx - cot x22 27. 2 sin2 x + 3 sinx + 1 28. 4 tan2 b + tanb - 3 29. cos4 x + 2 cos2 x + 1 30. cot4 x + 3 cot2 x + 2 31. sin3 x - cos3 x 32. sin3 a + cos3 a Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression. 33. tanu cos u 34. cot asina 35. sec r cos r 36. cot t tan t 37. sinbtanb cos b 38. csc u sec u cot u 39. sec2 x - 1 40. csc2 t - 1 41. sin2 x cos2 x + sinx csc x 42. 1 tan2 a + cot atana 43. 1 - 1 csc2 x 44. 1 - 1 sec2 x
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