## Solutions to Try Its

1. $\begin{array}{l}\csc \theta \cos \theta \tan \theta =\left(\frac{1}{\sin \theta }\right)\cos \theta \left(\frac{\sin \theta }{\cos \theta }\right)\hfill \\ \text{ }=\frac{\cos \theta }{\sin \theta }\left(\frac{\sin \theta }{\cos \theta }\right)\hfill \\ \text{ }=\frac{\sin \theta \cos \theta }{\sin \theta \cos \theta }\hfill \\ \text{ }=1\hfill \end{array}$

2. $\begin{array}{l}\frac{\cot \theta }{\csc \theta }=\frac{\frac{\cos \theta }{\sin \theta }}{\frac{1}{\sin \theta }}\hfill \\ \text{ }=\frac{\cos \theta }{\sin \theta }\cdot \frac{\sin \theta }{1}\hfill \\ \text{ }=\cos \theta \hfill \end{array}$

3. $\begin{array}{c}\frac{{\sin }^{2}\theta -1}{\tan \theta \sin \theta -\tan \theta }=\frac{\left(\sin \theta +1\right)\left(\sin \theta -1\right)}{\tan \theta \left(\sin \theta -1\right)}\\ =\frac{\sin \theta +1}{\tan \theta }\end{array}$

4. This is a difference of squares formula: $25 - 9{\sin }^{2}\theta =\left(5 - 3\sin \theta \right)\left(5+3\sin \theta \right)$.

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

## Solutions to Odd-Numbered Exercises

1. All three functions, $F,G$, and $H$, are even.

This is because $F\left(-x\right)=\sin \left(-x\right)\sin \left(-x\right)=\left(-\sin x\right)\left(-\sin x\right)={\sin }^{2}x=F\left(x\right),G\left(-x\right)=\cos \left(-x\right)\cos \left(-x\right)=\cos x\cos x={\cos }^{2}x=G\left(x\right)$ and $H\left(-x\right)=\tan \left(-x\right)\tan \left(-x\right)=\left(-\tan x\right)\left(-\tan x\right)={\tan }^{2}x=H\left(x\right)$.

3. When $\cos t=0$, then $\sec t=\frac{1}{0}$, which is undefined.

5. $\sin x$

7. $\sec x$

9. $\csc t$

11. $-1$

13. ${\sec }^{2}x$

15. ${\sin }^{2}x+1$

17. $\frac{1}{\sin x}$

19. $\frac{1}{\cot x}$

21. $\tan x$

23. $-4\sec x\tan x$

25. $\pm \sqrt{\frac{1}{{\cot }^{2}x}+1}$

27. $\frac{\pm \sqrt{1-{\sin }^{2}x}}{\sin x}$

29. Answers will vary. Sample proof:
$\cos x-{\cos }^{3}x=\cos x\left(1-{\cos }^{2}x\right)$

$=\cos x{\sin }^{2}x$

31. Answers will vary. Sample proof:

$\frac{1+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}+\frac{{\sin }^{2}x}{{\cos }^{2}x}={\sec }^{2}x+{\tan }^{2}x={\tan }^{2}x+1+{\tan }^{2}x=1+2{\tan }^{2}x$

33. Answers will vary. Sample proof:

${\cos }^{2}x-{\tan }^{2}x=1-{\sin }^{2}x-\left({\sec }^{2}x - 1\right)=1-{\sin }^{2}x-{\sec }^{2}x+1=2-{\sin }^{2}x-{\sec }^{2}x$

35. False

37. False

39. Proved with negative and Pythagorean identities

41. True

$3{\sin }^{2}\theta +4{\cos }^{2}\theta =3{\sin }^{2}\theta +3{\cos }^{2}\theta +{\cos }^{2}\theta =3\left({\sin }^{2}\theta +{\cos }^{2}\theta \right)+{\cos }^{2}\theta =3+{\cos }^{2}\theta$