Solutions 51: Solving Trigonometric Equations with Identities

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. ±1sin2xsinx

29. Answers will vary. Sample proof:
cosxcos3x=cosx(1cos2x)
=cosxsin2x

31. Answers will vary. Sample proof:

1+sin2xcos2x=1cos2x+sin2xcos2x=sec2x+tan2x=tan2x+1+tan2x=1+2tan2x

33. Answers will vary. Sample proof:

cos2xtan2x=1sin2x(sec2x1)=1sin2xsec2x+1=2sin2xsec2x

35. False

37. False

39. Proved with negative and Pythagorean identities

41. True

3sin2θ+4cos2θ=3sin2θ+3cos2θ+cos2θ=3(sin2θ+cos2θ)+cos2θ=3+cos2θ