Solution to Odd-Numbered Exercises

1. Use the Pythagorean identities and isolate the squared term.

3. $\frac{1-\cos x}{\sin x},\frac{\sin x}{1+\cos x}$, multiplying the top and bottom by $\sqrt{1-\cos x}$ and $\sqrt{1+\cos x}$, respectively.

5. a) $\frac{3\sqrt{7}}{32}$ b) $\frac{31}{32}$ c) $\frac{3\sqrt{7}}{31}$

7. a) $\frac{\sqrt{3}}{2}$ b) $-\frac{1}{2}$ c) $-\sqrt{3}$

9. $\cos \theta =-\frac{2\sqrt{5}}{5},\sin \theta =\frac{\sqrt{5}}{5},\tan \theta =-\frac{1}{2},\csc \theta =\sqrt{5},\sec \theta =-\frac{\sqrt{5}}{2},\cot \theta =-2$

11. $2\sin \left(\frac{\pi }{2}\right)$

13. $\frac{\sqrt{2-\sqrt{2}}}{2}$

15. $\frac{\sqrt{2-\sqrt{3}}}{2}$

17. $2+\sqrt{3}$

19. $-1-\sqrt{2}$

21. a) $\frac{3\sqrt{13}}{13}$ b) $-\frac{2\sqrt{13}}{13}$ c) $-\frac{3}{2}$

23. a) $\frac{\sqrt{10}}{4}$ b) $\frac{\sqrt{6}}{4}$ c) $\frac{\sqrt{15}}{3}$

25. $\frac{120}{169},-\frac{119}{169},-\frac{120}{119}$

27. $\frac{2\sqrt{13}}{13},\frac{3\sqrt{13}}{13},\frac{2}{3}$

29. $\cos \left({74}^{\circ }\right)$

31. $\cos \left(18x\right)$

33. $3\sin \left(10x\right)$

35. $-2\sin \left(-x\right)\cos \left(-x\right)=-2\left(-\sin \left(x\right)\cos \left(x\right)\right)=\sin \left(2x\right)$

37. $\begin{array}{l}\frac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)}{\tan }^{2}\theta =\frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{1+{\cos }^{2}\theta -{\sin }^{2}\theta }{\tan }^{2}\theta =\\ \frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{2{\cos }^{2}\theta }{\tan }^{2}\theta =\frac{\sin \left(\theta \right)}{\cos \theta }{\tan }^{2}\theta =\\ \cot \left(\theta \right){\tan }^{2}\theta =\tan \theta \end{array}$

39. $\frac{1+\cos \left(12x\right)}{2}$

41. $\frac{3+\cos \left(12x\right)-4\cos \left(6x\right)}{8}$

43. $\frac{2+\cos \left(2x\right)-2\cos \left(4x\right)-\cos \left(6x\right)}{32}$

45. $\frac{3+\cos \left(4x\right)-4\cos \left(2x\right)}{3+\cos \left(4x\right)+4\cos \left(2x\right)}$

47. $\frac{1-\cos \left(4x\right)}{8}$

49. $\frac{3+\cos \left(4x\right)-4\cos \left(2x\right)}{4\left(\cos \left(2x\right)+1\right)}$

51. $\frac{\left(1+\cos \left(4x\right)\right)\sin x}{2}$

53. $4\sin x\cos x\left({\cos }^{2}x-{\sin }^{2}x\right)$

55. $\frac{2\tan x}{1+{\tan }^{2}x}=\frac{\frac{2\sin x}{\cos x}}{1+\frac{{\sin }^{2}x}{{\cos }^{2}x}}=\frac{\frac{2\sin x}{\cos x}}{\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}}=$
$\frac{2\sin x}{\cos x}.\frac{{\cos }^{2}x}{1}=2\sin x\cos x=\sin \left(2x\right)$

57. $\frac{2\sin x\cos x}{2{\cos }^{2}x - 1}=\frac{\sin \left(2x\right)}{\cos \left(2x\right)}=\tan \left(2x\right)$

59. $\begin{array}{l}\sin \left(x+2x\right)=\sin x\cos \left(2x\right)+\sin \left(2x\right)\cos x\hfill \\ =\sin x\left({\cos }^{2}x-{\sin }^{2}x\right)+2\sin x\cos x\cos x\hfill \\ =\sin x{\cos }^{2}x-{\sin }^{3}x+2\sin x{\cos }^{2}x\hfill \\ =3\sin x{\cos }^{2}x-{\sin }^{3}x\hfill \end{array}$

61. $\begin{array}{l}\frac{1+\cos \left(2t\right)}{\sin \left(2t\right)-\cos t}=\frac{1+2{\cos }^{2}t - 1}{2\sin t\cos t-\cos t}\hfill \\ =\frac{2{\cos }^{2}t}{\cos t\left(2\sin t - 1\right)}\hfill \\ =\frac{2\cos t}{2\sin t - 1}\hfill \end{array}$

63. $\begin{array}{l}\left({\cos }^{2}\left(4x\right)-{\sin }^{2}\left(4x\right)-\sin \left(8x\right)\right)\left({\cos }^{2}\left(4x\right)-{\sin }^{2}\left(4x\right)+\sin \left(8x\right)\right)=\hfill \\ \text{ }=\left(\cos \left(8x\right)-\sin \left(8x\right)\right)\left(\cos \left(8x\right)+\sin \left(8x\right)\right)\hfill \\ \text{ }={\cos }^{2}\left(8x\right)-{\sin }^{2}\left(8x\right)\hfill \\ \text{ }=\cos \left(16x\right)\hfill \\ \hfill \end{array}$