Practice and Problem-Solving Exercises

A Practice

Verify each identity. Give the domain of validity for each identity. See Problems 1–4.

7. cos θ cot θ = 1 sin θ − sin θ cosine theta co-tangent theta equals . fraction 1 , over sine theta end fraction . minus sine theta
8. sin θ cot θ = cos θ sine theta co-tangent theta equals cosine theta
9. cos θ tan θ = sin θ cosine theta tangent theta equals sine theta
10. sin θ sec θ = tan θ sine theta secant theta equals tangent theta
11. cos θ sec θ = 1 cosine theta secant theta equals 1
12. tan θ cot θ = 1 tangent theta co-tangent theta equals 1
13. sin θ  csc  θ = 1 sine theta , csc , theta equals 1
14. cot θ =  csc  θ cos θ co-tangent theta equals , csc , theta cosine theta
15. csc θ − sin θ = cot θ cos θ co-secant theta negative sine theta equals co-tangent theta cosine theta

Simplify each trigonometric expression. See Problem 5.

16. tan θ cot θ tangent theta co-tangent theta
17. 1 − cos 2 θ 1 minus , cosine squared , theta
18. sec 2 θ − 1 secant squared , theta negative 1
19. 1 − csc 2 θ 1 minus co-secant 2 theta
20. sec 2 θ cot 2 θ secant squared , theta , co-tangent squared , theta
21. cos θ tan θ cosine theta tangent theta
22. sin θ cot θ sine theta co-tangent theta
23. sin θ csc θ sine theta co-secant theta
24. sec θ cos θ sin θ secant theta cosine theta sine theta
25. sin θ sec θ cot θ sine theta secant theta co-tangent theta
26. sec 2 θ − tan 2 θ secant squared , theta negative , tangent squared , theta
27. sin θ cos θ tan θ fraction sine theta , over cosine theta tangent theta end fraction

B Apply

28. Think About a Plan Simplify the expression tan θ sec θ − cos θ . fraction tangent theta , over secant theta minus cosine theta end fraction . .

    • Can you write everything in terms of sin θ , cos θ , sine theta comma . cosine theta comma or both?
    • Are there any trigonometric identities that can help you simplify the expression?

Simplify each trigonometric expression.

29. cos θ + sin θ tan θ cosine theta plus sine theta tangent theta
30. csc θ cos θ tan θ co-secant theta cosine theta tangent theta
31. tan θ ( cot θ + tan θ ) tangent theta open co-tangent theta plus tangent theta close
32. sin 2 θ + cos 2 θ + tan 2 θ sine squared , theta plus , cosine squared , theta plus , tangent squared , theta
33. sin θ ( 1 + cot 2 θ ) sine theta open 1 plus , co-tangent squared , theta close
34. sin 2 θ csc θ sec θ sine squared , theta co-secant theta secant theta
35. sec θ cos θ − cos 2 θ secant theta cosine theta negative , cosine squared , theta
36. csc θ − cos θ cot θ co-secant theta negative cosine theta co-tangent theta
37. csc 2 θ ( 1 − cos 2 θ ) co-secant squared , theta open 1 minus , cosine squared , theta close
38. csc θ sin θ + cos θ cot θ fraction co-secant theta , over sine theta plus cosine theta co-tangent theta end fraction
39. cos θ csc θ cot θ fraction cosine theta co-secant theta , over co-tangent theta end fraction
40. sin 2 θ csc θ sec θ tan θ fraction sine squared , theta co-secant theta secant theta , over tangent theta end fraction

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