Problem Set 51: Solving Trigonometric Equations With Identities

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

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

3. After examining the reciprocal identity for [latex]\sec t[/latex], 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 [latex]{\sin }^{2}t+{\cos }^{2}t=1[/latex] to the other forms.

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

5. [latex]\sin x\cos x\sec x[/latex]

6. [latex]\sin \left(-x\right)\cos \left(-x\right)\csc \left(-x\right)[/latex]

7. [latex]\tan x\sin x+\sec x{\cos }^{2}x[/latex]

8. [latex]\csc x+\cos x\cot \left(-x\right)[/latex]

9. [latex]\frac{\cot t+\tan t}{\sec \left(-t\right)}[/latex]

10. [latex]3{\sin }^{3}t\csc t+{\cos }^{2}t+2\cos \left(-t\right)\cos t[/latex]

11. [latex]-\tan \left(-x\right)\cot \left(-x\right)[/latex]

12. [latex]\frac{-\sin \left(-x\right)\cos x\sec x\csc x\tan x}{\cot x}[/latex]

13. [latex]\frac{1+{\tan }^{2}\theta }{{\csc }^{2}\theta }+{\sin }^{2}\theta +\frac{1}{{\sec }^{2}\theta }[/latex]

14. [latex]\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}[/latex]

15. [latex]\frac{1-{\cos }^{2}x}{{\tan }^{2}x}+2{\sin }^{2}x[/latex]

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

16. [latex]\frac{\tan x+\cot x}{\csc x};\cos x[/latex]

17. [latex]\frac{\sec x+\csc x}{1+\tan x};\sin x[/latex]

18. [latex]\frac{\cos x}{1+\sin x}+\tan x;\cos x[/latex]

19. [latex]\frac{1}{\sin x\cos x}-\cot x;\cot x[/latex]

20. [latex]\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x};\csc x[/latex]

21. [latex]\left(\sec x+\csc x\right)\left(\sin x+\cos x\right)-2-\cot x;\tan x[/latex]

22. [latex]\frac{1}{\csc x-\sin x};\sec x\text{ and }\tan x[/latex]

23. [latex]\frac{1-\sin x}{1+\sin x}-\frac{1+\sin x}{1-\sin x};\sec x\text{ and }\tan x[/latex]

24. [latex]\tan x;\sec x[/latex]

25. [latex]\sec x;\cot x[/latex]

26. [latex]\sec x;\sin x[/latex]

27. [latex]\cot x;\sin x[/latex]

28. [latex]\cot x;\csc x[/latex]

For the following exercises, verify the identity.

29. [latex]\cos x-{\cos }^{3}x=\cos x{\sin }^{2}x[/latex]

30. [latex]\cos x\left(\tan x-\sec \left(-x\right)\right)=\sin x - 1[/latex]

31. [latex]\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[/latex]

32. [latex]{\left(\sin x+\cos x\right)}^{2}=1+2\sin x\cos x[/latex]

33. [latex]{\cos }^{2}x-{\tan }^{2}x=2-{\sin }^{2}x-{\sec }^{2}x[/latex]

For the following exercises, prove or disprove the identity.

34. [latex]\frac{1}{1+\cos x}-\frac{1}{1-\cos \left(-x\right)}=-2\cot x\csc x[/latex]

35. [latex]{\csc }^{2}x\left(1+{\sin }^{2}x\right)={\cot }^{2}x[/latex]

36. [latex]\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[/latex]

37. [latex]\frac{\tan x}{\sec x}\sin \left(-x\right)={\cos }^{2}x[/latex]

38. [latex]\frac{\sec \left(-x\right)}{\tan x+\cot x}=-\sin \left(-x\right)[/latex]

39. [latex]\frac{1+\sin x}{\cos x}=\frac{\cos x}{1+\sin \left(-x\right)}[/latex]

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

40. [latex]\frac{{\cos }^{2}\theta -{\sin }^{2}\theta }{1-{\tan }^{2}\theta }={\sin }^{2}\theta [/latex]

41. [latex]3{\sin }^{2}\theta +4{\cos }^{2}\theta =3+{\cos }^{2}\theta [/latex]

42. [latex]\frac{\sec \theta +\tan \theta }{\cot \theta +\cos \theta }={\sec }^{2}\theta [/latex]