# 9.1E: Solving Trigonometric Equations with Identities (Exercises)

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For the following exercises, find all solutions exactly that exist on the interval $$[0,2 \pi)$$.

1. $$\csc ^{2} t=3$$

2. $$\cos ^{2} x=\frac{1}{4}$$

3. $$2 \sin \theta=-1$$

4. $$\tan x \sin x+\sin (-x)=0$$

5. $$9 \sin \omega-2=4 \sin ^{2} \omega$$

6. $$1-2 \tan (\omega)=\tan ^{2}(\omega)$$

For the following exercises, use basic identities to simplify the expression.

7. $$\sec x \cos x+\cos x-\frac{1}{\sec x}$$

8. $$\sin ^{3} x+\cos ^{2} x \sin x$$

For the following exercises, determine if the given identities are equivalent.

9. $$\sin ^{2} x+\sec ^{2} x-1=\frac{\left(1-\cos ^{2} x\right)\left(1+\cos ^{2} x\right)}{\cos ^{2} x}$$

10. $$\tan ^{3} x \csc ^{2} x \cot ^{2} x \cos x \sin x=1$$

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