Trig identities - stuck solving $\tan^2\theta = -\frac 32 \sec\theta$ Solve the equation on the interval $0\leq \theta < 2\pi$
$$\tan^2 \theta = -\frac{3}{2}\sec \theta $$
Here are the steps I have so far:
Identity: $\tan^2 \theta = \sec^2 \theta -1 $
Substitute:
$$\sec^2 \theta -1  = -\frac{3}{2}\sec \theta $$
$$2\sec^2 \theta -2 = {-3}\sec \theta $$
$$2\sec^2 \theta +3\sec \theta - 2 = 0 $$
Is this factoring correct?:
$$(2\sec\theta+4)(\sec\theta-1) = 0 $$
$$2\sec\theta+4 =0$$
$$2\sec\theta = -4 $$
$$\sec \theta = -2$$
$$(2\pi/3), (4\pi/3) $$
$$\sec\theta - 1 = 0$$
$$\sec\theta = 1 $$
$\sec\theta=1$ would evaluate to $0$, for some reason that is an invalid answer? (according to my assignment)
So is $\ (2\pi/3), (4\pi/3) $ the full answer? 

No, the factoring is not correct - it should be $\left(2\sec\theta +4\right)\left(\sec\theta- \frac12\right)$
