Express the first trigonometric function in terms of the second.

41. sin θ , cos θ sine theta comma cosine theta
42. tan θ , cos θ tangent theta comma cosine theta
43. cot θ , sin θ co-tangent theta comma sine theta
44. csc θ , cot θ co-secant theta comma co-tangent theta
45. cot θ , csc θ co-tangent theta comma co-secant theta
46. sec θ , tan θ secant theta comma tangent theta

Verify each identity.

47. sin 2 θ tan 2 θ = tan 2 θ − sin 2 θ sine squared , theta , tangent squared , theta equals , tangent squared , theta negative , sine squared , theta
48. sec θ − sin θ tan θ = cos θ secant theta negative sine theta tangent theta equals cosine theta
49. sin θ cos θ ( tan θ + cot θ ) = 1 sine theta cosine theta open tangent theta plus co-tangent theta close equals 1
50. 1 − sin θ cos θ = cos θ 1 + sin θ fraction 1 minus sine theta , over cosine theta end fraction . equals . fraction cosine theta , over 1 plus sine theta end fraction
51. sec θ cot θ + tan θ = sin θ fraction secant theta , over co-tangent theta plus tangent theta end fraction . equals sine theta
52. ( cot θ + 1 ) 2 = csc 2 θ + 2 cot θ open co-tangent theta plus 1 close squared . equals , co-secant squared , theta plus 2 co-tangent theta
53. Express cos θ csc θ cot θ cosine theta co-secant theta co-tangent theta in terms of sin θ . sine theta .
54. Express cos θ sec θ + tan θ fraction cosine theta , over secant theta plus tangent theta end fraction in terms of sin θ . sine theta .

55. Error Analysis Find the two errors in the verification of the identity sec 2 θ − tan 2 θ tan 2 θ = cot 2 θ fraction secant squared , theta minus , tangent squared , theta , over tangent squared , theta end fraction . equals , co-tangent squared , theta shown below. Then verify the identity correctly.

    
    Image Long Description

56. Open-Ended Develop your own trigonometric identity. (Hint: Start with a simple trigonometric expression and work backward.)
57. Writing Only one of the following equations is an identity. Identify the identity and explain your answer.
    • ( x − 1 ) 2 − 1 = x ( x − 2 ) ( x − 1 ) 2 = x ( x − 1 ) open x minus 1 close squared . minus 1 equals x open x minus 2 close . open x minus 1 close squared . equals x open x minus 1 close

C Challenge

58. The unit circle is a useful tool for verifying identities. Use the diagram below to verify the identity sin ( θ + π ) = − sin θ . sine open theta plus pi close equals negative sine theta .

    

    1. Explain why the y-coordinate of point P is sin ( θ + π ) . sine open theta plus . pi close .
    2. Prove that the two triangles shown are congruent.
    3. Use part (b) to show that the two blue segments are congruent.
    4. Use part (c) to show that the y-coordinate of P is − sin θ . negative sine theta .
    5. Use parts (a) and (d) to conclude that sin ( θ + π ) = − sin θ . sine open theta plus pi close equals negative sine theta .

Use the diagram in Exercise 58 to verify each identity.

59. cos ( θ + π ) = − cos θ cosine open theta plus pi close equals negative cosine theta
60. tan ( θ + π ) = tan θ tangent open theta plus pi close equals tangent theta

Simplify each trigonometric expression.

61. cot 2 θ − csc 2 θ tan 2 θ − sec 2 θ fraction co-tangent squared , theta minus , co-secant squared , theta , over tangent squared , theta minus , secant squared , theta end fraction
62. ( 1 − sin θ ) ( 1 + sin θ ) csc 2 θ + 1 open 1 minus sine theta close open 1 plus sine theta close , co-secant squared , theta plus 1

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