## Problem Set 51: Solving Trigonometric Equations With Identities

1. We know $g\left(x\right)=\cos x$ is an even function, and $f\left(x\right)=\sin x$ and $h\left(x\right)=\tan x$ are odd functions. What about $G\left(x\right)={\cos }^{2}x,F\left(x\right)={\sin }^{2}x$, and $H\left(x\right)={\tan }^{2}x?$ Are they even, odd, or neither? Why?

2. Examine the graph of $f\left(x\right)=\sec x$ on the interval $\left[-\pi ,\pi \right]$. How can we tell whether the function is even or odd by only observing the graph of $f\left(x\right)=\sec x?$

3. After examining the reciprocal identity for $\sec t$, explain why the function is undefined at certain points.

4. All of the Pythagorean identities are related. Describe how to manipulate the equations to get from ${\sin }^{2}t+{\cos }^{2}t=1$ to the other forms.

For the following exercises, use the fundamental identities to fully simplify the expression.

5. $\sin x\cos x\sec x$

6. $\sin \left(-x\right)\cos \left(-x\right)\csc \left(-x\right)$

7. $\tan x\sin x+\sec x{\cos }^{2}x$

8. $\csc x+\cos x\cot \left(-x\right)$

9. $\frac{\cot t+\tan t}{\sec \left(-t\right)}$

10. $3{\sin }^{3}t\csc t+{\cos }^{2}t+2\cos \left(-t\right)\cos t$

11. $-\tan \left(-x\right)\cot \left(-x\right)$

12. $\frac{-\sin \left(-x\right)\cos x\sec x\csc x\tan x}{\cot x}$

13. $\frac{1+{\tan }^{2}\theta }{{\csc }^{2}\theta }+{\sin }^{2}\theta +\frac{1}{{\sec }^{2}\theta }$

14. $\left(\frac{\tan x}{{\csc }^{2}x}+\frac{\tan x}{{\sec }^{2}x}\right)\left(\frac{1+\tan x}{1+\cot x}\right)-\frac{1}{{\cos }^{2}x}$

15. $\frac{1-{\cos }^{2}x}{{\tan }^{2}x}+2{\sin }^{2}x$

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

16. $\frac{\tan x+\cot x}{\csc x};\cos x$

17. $\frac{\sec x+\csc x}{1+\tan x};\sin x$

18. $\frac{\cos x}{1+\sin x}+\tan x;\cos x$

19. $\frac{1}{\sin x\cos x}-\cot x;\cot x$

20. $\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x};\csc x$

21. $\left(\sec x+\csc x\right)\left(\sin x+\cos x\right)-2-\cot x;\tan x$

22. $\frac{1}{\csc x-\sin x};\sec x\text{ and }\tan x$

23. $\frac{1-\sin x}{1+\sin x}-\frac{1+\sin x}{1-\sin x};\sec x\text{ and }\tan x$

24. $\tan x;\sec x$

25. $\sec x;\cot x$

26. $\sec x;\sin x$

27. $\cot x;\sin x$

28. $\cot x;\csc x$

For the following exercises, verify the identity.

29. $\cos x-{\cos }^{3}x=\cos x{\sin }^{2}x$

30. $\cos x\left(\tan x-\sec \left(-x\right)\right)=\sin x - 1$

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

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

33. ${\cos }^{2}x-{\tan }^{2}x=2-{\sin }^{2}x-{\sec }^{2}x$

For the following exercises, prove or disprove the identity.

34. $\frac{1}{1+\cos x}-\frac{1}{1-\cos \left(-x\right)}=-2\cot x\csc x$

35. ${\csc }^{2}x\left(1+{\sin }^{2}x\right)={\cot }^{2}x$

36. $\left(\frac{{\sec }^{2}\left(-x\right)-{\tan }^{2}x}{\tan x}\right)\left(\frac{2+2\tan x}{2+2\cot x}\right)-2{\sin }^{2}x=\cos 2x$

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

38. $\frac{\sec \left(-x\right)}{\tan x+\cot x}=-\sin \left(-x\right)$

39. $\frac{1+\sin x}{\cos x}=\frac{\cos x}{1+\sin \left(-x\right)}$

For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.

40. $\frac{{\cos }^{2}\theta -{\sin }^{2}\theta }{1-{\tan }^{2}\theta }={\sin }^{2}\theta$

41. $3{\sin }^{2}\theta +4{\cos }^{2}\theta =3+{\cos }^{2}\theta$

42. $\frac{\sec \theta +\tan \theta }{\cot \theta +\cos \theta }={\sec }^{2}\theta$