9.1E: Solving Trigonometric Equations with Identities (Exercises)
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. \(\tan ^{3} x \csc ^{2} x \cot ^{2} x \cos x \sin x=1\)