Solutions for Solving Trigonometric Equations with Identities

Solutions to Try Its

1. [latex]\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}[/latex]

2. [latex]\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}[/latex]

3. [latex]\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}[/latex]

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

5. [latex]\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}[/latex]

Solutions to Odd-Numbered Exercises

1. All three functions, [latex]F,G[/latex], and [latex]H[/latex], are even.

This is because [latex]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)[/latex] and [latex]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)[/latex].

3. When [latex]\cos t=0[/latex], then [latex]\sec t=\frac{1}{0}[/latex], which is undefined.

5. [latex]\sin x[/latex]

7. [latex]\sec x[/latex]

9. [latex]\csc t[/latex]

11. [latex]-1[/latex]

13. [latex]{\sec }^{2}x[/latex]

15. [latex]{\sin }^{2}x+1[/latex]

17. [latex]\frac{1}{\sin x}[/latex]

19. [latex]\frac{1}{\cot x}[/latex]

21. [latex]\tan x[/latex]

23. [latex]-4\sec x\tan x[/latex]

25. [latex]\pm \sqrt{\frac{1}{{\cot }^{2}x}+1}[/latex]

27. [latex]\frac{\pm \sqrt{1-{\sin }^{2}x}}{\sin x}[/latex]

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

[latex]=\cos x{\sin }^{2}x[/latex]

31. Answers will vary. Sample proof:

[latex]\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[/latex]

33. Answers will vary. Sample proof:

[latex]{\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[/latex]

35. False

37. False

39. Proved with negative and Pythagorean identities

41. True

[latex]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 [/latex]