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# 9.3E: Double-Angle, Half-Angle, and Reduction Formulas (Exercises)

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For the following exercises, find the exact value.

20. Find $$\sin (2 \theta), \cos (2 \theta),$$ and $$\tan (2 \theta)$$ given $$\cos \theta=-\frac{1}{3}$$ and $$\theta$$ is in the interval $$\left[\frac{\pi}{2}, \pi\right]$$.

21. Find $$\sin (2 \theta), \cos (2 \theta),$$ and $$\tan (2 \theta)$$ given sec $$\theta=-\frac{5}{3}$$ and $$\theta$$ is in the interval $$\left[\frac{\pi}{2}, \pi\right]$$.

22. $$\sin \left(\frac{7 \pi}{8}\right)$$

23. $$\sec \left(\frac{3 \pi}{8}\right)$$

For the following exercises, use Figure 1 to find the desired quantities. Figure 1

24. $$\sin (2 \beta), \cos (2 \beta), \tan (2 \beta), \sin (2 \alpha), \cos (2 \alpha),$$ and $$\tan (2 \alpha)$$

25. $$\sin \left(\frac{\beta}{2}\right), \cos \left(\frac{\beta}{2}\right), \tan \left(\frac{\beta}{2}\right), \sin \left(\frac{\alpha}{2}\right), \cos \left(\frac{\alpha}{2}\right),$$ and $$\tan \left(\frac{\alpha}{2}\right)$$

For the following exercises, prove the identity.

26. $$\frac{2 \cos (2 x)}{\sin (2 x)}=\cot x-\tan x$$

27. $$\cot x \cos (2 x)=-\sin (2 x)+\cot x$$

For the following exercises, rewrite the expression with no powers.

28. $$\cos ^{2} x \sin ^{4}(2 x)$$

29. $$\tan ^{2} x \sin ^{3} x$$