Answer: \(\theta\approx -0.644\) or \(\theta\approx0.464\).

The key point is to reduce the equation to a standard trigonometric equation, then use the given interval to list all valid solutions.

Use the identity

\(\operatorname{cosec}^2\theta=1+\operatorname{cot}^2\theta.\)

The equation becomes

\(1+\operatorname{cot}^2\theta+2\operatorname{cot}^2\theta=2\operatorname{cot}\theta+9.\)

So

\(3\operatorname{cot}^2\theta-2\operatorname{cot}\theta-8=0.\)

Factorise:

\((3\operatorname{cot}\theta+4)(\operatorname{cot}\theta-2)=0.\)

Thus

Equivalently,

Since

\(-\frac{\pi}{2}\lt \theta\lt \frac{\pi}{2},\)

there is one solution for each tangent value: