Algebra and Trigonometry, 5th Edition

Mid-Chapter Mixed Review

Determine whether the statement is true or false.

1. $\text{sin}x\left(\text{csc}x-\text{cot}x\right)=1-\text{cos}x$ [7.1]

2. $\text{sin}42°=\sqrt{{\displaystyle \frac{1+\text{cos}84°}{2}}}$ [7.2]

3. $\text{sin}{\displaystyle \frac{\pi }{9}}=\text{cos}{\displaystyle \frac{7\pi }{18}}$ [7.2]

4. ${\text{cos}}^{2}x\ne \text{cos}{x}^2$ [7.1]

For Exercises 5–14, choose one of expressions A–J to complete the identity. [7.1], [7.2]

5. $\text{cos}\left(-x\right)=$

6. $\text{cos}\left(u+v\right)=$

7. $\text{tan}2x=$

8. $\text{tan}\left({\displaystyle \frac{\pi }{2}}-x\right)=$

9. $1+{\text{cot}}^{2}x=$

10. $\text{sin}{\displaystyle \frac{x}{2}}=$

11. $\text{sin}2x=$

12. $\text{sin}\left(u-v\right)=$

13. $\text{csc}\left({\displaystyle \frac{\pi }{2}}-x\right)=$

14. $\text{cos}{\displaystyle \frac{x}{2}}=$

    1. $2\text{sin}x\text{cos}x$

    2. $\pm \sqrt{{\displaystyle \frac{1+\text{cos}x}{2}}}$

    3. ${\text{csc}}^{2}x$

    4. ${\displaystyle \frac{2\text{tan}x}{1-{\text{tan}}^{2}x}}$

    5. $\pm \sqrt{{\displaystyle \frac{1-\text{cos}x}{2}}}$

    6. sec x

    7. $\text{sin}u\text{cos}v-\text{cos}u\text{sin}v$

    8. $\text{cos}u\text{cos}v-\text{sin}u\text{sin}v$

    9. cot x

    10. cos x

Simplify.

15. $\sqrt{{\displaystyle \frac{\text{cot}x}{\text{sin}x}}}$ [7.1]

16. ${\displaystyle \frac{1}{{\text{sin}}^{2}x}}-{\left({\displaystyle \frac{\text{cos}x}{\text{sin}x}}\right)}^2$ [7.1]

17. ${\displaystyle \frac{2{\text{cos}}^{2}x-5\text{cos}x-3}{\text{cos}x-3}}$ [7.1]

18. ${\displaystyle \frac{\text{sin}x}{\text{tan}\left(-x\right)}}$ [7.1]

19. ${\left(\text{cos}x-\text{sin}x\right)}^2$ [7.2]

20. $1-2{\text{sin}}^2{\displaystyle \frac{x}{2}}$ [7.2]

21. Rationalize the denominator: $\sqrt{{\displaystyle \frac{\text{sec}x}{1-\text{cos}x}}}.$ [7.1]

22. Write $\text{cos}41°\text{cos}$