{"id":15687,"date":"2019-09-05T17:47:05","date_gmt":"2019-09-05T17:47:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&p=15687"},"modified":"2025-02-05T05:23:31","modified_gmt":"2025-02-05T05:23:31","slug":"problem-set-51-solving-trigonometric-equations-with-identities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-51-solving-trigonometric-equations-with-identities\/","title":{"raw":"Problem Set 51: Solving Trigonometric Equations With Identities","rendered":"Problem Set 51: Solving Trigonometric Equations With Identities"},"content":{"raw":"
<\/section>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?\r\n\r\n2. 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]\r\n\r\n3. After examining the reciprocal identity for [latex]\\sec t[\/latex], explain why the function is undefined at certain points.\r\n\r\n4. 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.\r\n\r\nFor the following exercises, use the fundamental identities to fully simplify the expression.\r\n\r\n5. [latex]\\sin x\\cos x\\sec x[\/latex]\r\n\r\n6.\u00a0[latex]\\sin \\left(-x\\right)\\cos \\left(-x\\right)\\csc \\left(-x\\right)[\/latex]\r\n\r\n7. [latex]\\tan x\\sin x+\\sec x{\\cos }^{2}x[\/latex]\r\n\r\n8.\u00a0[latex]\\csc x+\\cos x\\cot \\left(-x\\right)[\/latex]\r\n\r\n9.\u00a0[latex]\\frac{\\cot t+\\tan t}{\\sec \\left(-t\\right)}[\/latex]\r\n\r\n10.\u00a0[latex]3{\\sin }^{3}t\\csc t+{\\cos }^{2}t+2\\cos \\left(-t\\right)\\cos t[\/latex]\r\n\r\n11. [latex]-\\tan \\left(-x\\right)\\cot \\left(-x\\right)[\/latex]\r\n\r\n12.\u00a0[latex]\\frac{-\\sin \\left(-x\\right)\\cos x\\sec x\\csc x\\tan x}{\\cot x}[\/latex]\r\n\r\n13. [latex]\\frac{1+{\\tan }^{2}\\theta }{{\\csc }^{2}\\theta }+{\\sin }^{2}\\theta +\\frac{1}{{\\sec }^{2}\\theta }[\/latex]\r\n\r\n14.\u00a0[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]\r\n\r\n15. [latex]\\frac{1-{\\cos }^{2}x}{{\\tan }^{2}x}+2{\\sin }^{2}x[\/latex]\r\n\r\nFor the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.\r\n\r\n16. [latex]\\frac{\\tan x+\\cot x}{\\csc x};\\cos x[\/latex]\r\n\r\n17. [latex]\\frac{\\sec x+\\csc x}{1+\\tan x};\\sin x[\/latex]\r\n\r\n18.\u00a0[latex]\\frac{\\cos x}{1+\\sin x}+\\tan x;\\cos x[\/latex]\r\n\r\n19. [latex]\\frac{1}{\\sin x\\cos x}-\\cot x;\\cot x[\/latex]\r\n\r\n20.\u00a0[latex]\\frac{1}{1-\\cos x}-\\frac{\\cos x}{1+\\cos x};\\csc x[\/latex]\r\n\r\n21. [latex]\\left(\\sec x+\\csc x\\right)\\left(\\sin x+\\cos x\\right)-2-\\cot x;\\tan x[\/latex]\r\n\r\n22.\u00a0[latex]\\frac{1}{\\csc x-\\sin x};\\sec x\\text{ and }\\tan x[\/latex]\r\n\r\n23. [latex]\\frac{1-\\sin x}{1+\\sin x}-\\frac{1+\\sin x}{1-\\sin x};\\sec x\\text{ and }\\tan x[\/latex]\r\n\r\n24.\u00a0[latex]\\tan x;\\sec x[\/latex]\r\n\r\n25. [latex]\\sec x;\\cot x[\/latex]\r\n\r\n26.\u00a0[latex]\\sec x;\\sin x[\/latex]\r\n\r\n27. [latex]\\cot x;\\sin x[\/latex]\r\n\r\n28.\u00a0[latex]\\cot x;\\csc x[\/latex]\r\n\r\nFor the following exercises, verify the identity.\r\n\r\n29. [latex]\\cos x-{\\cos }^{3}x=\\cos x{\\sin }^{2}x[\/latex]\r\n\r\n30. [latex]\\cos x\\left(\\tan x-\\sec \\left(-x\\right)\\right)=\\sin x - 1[\/latex]\r\n\r\n31. [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]\r\n\r\n32. [latex]{\\left(\\sin x+\\cos x\\right)}^{2}=1+2\\sin x\\cos x[\/latex]\r\n\r\n33. [latex]{\\cos }^{2}x-{\\tan }^{2}x=2-{\\sin }^{2}x-{\\sec }^{2}x[\/latex]\r\n\r\nFor the following exercises, prove or disprove the identity.\r\n\r\n34. [latex]\\frac{1}{1+\\cos x}-\\frac{1}{1-\\cos \\left(-x\\right)}=-2\\cot x\\csc x[\/latex]\r\n\r\n35. [latex]{\\csc }^{2}x\\left(1+{\\sin }^{2}x\\right)={\\cot }^{2}x[\/latex]\r\n\r\n36.\u00a0[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]\r\n\r\n37. [latex]\\frac{\\tan x}{\\sec x}\\sin \\left(-x\\right)={\\cos }^{2}x[\/latex]\r\n\r\n38.\u00a0[latex]\\frac{\\sec \\left(-x\\right)}{\\tan x+\\cot x}=-\\sin \\left(-x\\right)[\/latex]\r\n\r\n39. [latex]\\frac{1+\\sin x}{\\cos x}=\\frac{\\cos x}{1+\\sin \\left(-x\\right)}[\/latex]\r\n\r\nFor the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.\r\n\r\n40. [latex]\\frac{{\\cos }^{2}\\theta -{\\sin }^{2}\\theta }{1-{\\tan }^{2}\\theta }={\\sin }^{2}\\theta [\/latex]\r\n\r\n41. [latex]3{\\sin }^{2}\\theta +4{\\cos }^{2}\\theta =3+{\\cos }^{2}\\theta [\/latex]\r\n\r\n42.\u00a0[latex]\\frac{\\sec \\theta +\\tan \\theta }{\\cot \\theta +\\cos \\theta }={\\sec }^{2}\\theta [\/latex]","rendered":"
<\/section>\n

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?<\/p>\n

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]<\/p>\n

3. After examining the reciprocal identity for [latex]\\sec t[\/latex], explain why the function is undefined at certain points.<\/p>\n

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.<\/p>\n

For the following exercises, use the fundamental identities to fully simplify the expression.<\/p>\n

5. [latex]\\sin x\\cos x\\sec x[\/latex]<\/p>\n

6.\u00a0[latex]\\sin \\left(-x\\right)\\cos \\left(-x\\right)\\csc \\left(-x\\right)[\/latex]<\/p>\n

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

8.\u00a0[latex]\\csc x+\\cos x\\cot \\left(-x\\right)[\/latex]<\/p>\n

9.\u00a0[latex]\\frac{\\cot t+\\tan t}{\\sec \\left(-t\\right)}[\/latex]<\/p>\n

10.\u00a0[latex]3{\\sin }^{3}t\\csc t+{\\cos }^{2}t+2\\cos \\left(-t\\right)\\cos t[\/latex]<\/p>\n

11. [latex]-\\tan \\left(-x\\right)\\cot \\left(-x\\right)[\/latex]<\/p>\n

12.\u00a0[latex]\\frac{-\\sin \\left(-x\\right)\\cos x\\sec x\\csc x\\tan x}{\\cot x}[\/latex]<\/p>\n

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

14.\u00a0[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]<\/p>\n

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

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.<\/p>\n

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

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

18.\u00a0[latex]\\frac{\\cos x}{1+\\sin x}+\\tan x;\\cos x[\/latex]<\/p>\n

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

20.\u00a0[latex]\\frac{1}{1-\\cos x}-\\frac{\\cos x}{1+\\cos x};\\csc x[\/latex]<\/p>\n

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

22.\u00a0[latex]\\frac{1}{\\csc x-\\sin x};\\sec x\\text{ and }\\tan x[\/latex]<\/p>\n

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

24.\u00a0[latex]\\tan x;\\sec x[\/latex]<\/p>\n

25. [latex]\\sec x;\\cot x[\/latex]<\/p>\n

26.\u00a0[latex]\\sec x;\\sin x[\/latex]<\/p>\n

27. [latex]\\cot x;\\sin x[\/latex]<\/p>\n

28.\u00a0[latex]\\cot x;\\csc x[\/latex]<\/p>\n

For the following exercises, verify the identity.<\/p>\n

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

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

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]<\/p>\n

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

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

For the following exercises, prove or disprove the identity.<\/p>\n

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

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

36.\u00a0[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]<\/p>\n

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

38.\u00a0[latex]\\frac{\\sec \\left(-x\\right)}{\\tan x+\\cot x}=-\\sin \\left(-x\\right)[\/latex]<\/p>\n

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

For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.<\/p>\n

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

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

42.\u00a0[latex]\\frac{\\sec \\theta +\\tan \\theta }{\\cot \\theta +\\cos \\theta }={\\sec }^{2}\\theta[\/latex]<\/p>\n\n\t\t\t

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