Solutions to Odd-Numbered Exercises
1. All three functions, F,G, and H, are even.
This is because F(−x)=sin(−x)sin(−x)=(−sinx)(−sinx)=sin2x=F(x),G(−x)=cos(−x)cos(−x)=cosxcosx=cos2x=G(x) and H(−x)=tan(−x)tan(−x)=(−tanx)(−tanx)=tan2x=H(x).
3. When cost=0, then sect=10, which is undefined.
5. sinx
7. secx
9. csct
11. −1
13. sec2x
15. sin2x+1
17. 1sinx
19. 1cotx
21. tanx
23. −4secxtanx
25. ±√1cot2x+1
27. ±√1−sin2xsinx
29. Answers will vary. Sample proof:
cosx−cos3x=cosx(1−cos2x)
=cosxsin2x
31. Answers will vary. Sample proof:
1+sin2xcos2x=1cos2x+sin2xcos2x=sec2x+tan2x=tan2x+1+tan2x=1+2tan2x
33. Answers will vary. Sample proof:
cos2x−tan2x=1−sin2x−(sec2x−1)=1−sin2x−sec2x+1=2−sin2x−sec2x
35. False
37. False
39. Proved with negative and Pythagorean identities
41. True
3sin2θ+4cos2θ=3sin2θ+3cos2θ+cos2θ=3(sin2θ+cos2θ)+cos2θ=3+cos2θ
Candela Citations
- Precalculus. Authored by: OpenStax College. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution