Solution

Find all solutions to the following equation

$$\sec \theta \sin^2 \theta = \tan \theta$$

Note that there is more than one trigonometric function involved in this equation. We prefer to have only one, as this generally makes solving things easier.

To reduce the number of trigonometric functions involved, let us translate the functions we have into expressions involving only sines and cosines, as the definitions for the remaining trigonometric functions allow,

$$\frac{1}{\cos \theta} \cdot \sin^2 \theta = \frac{\sin \theta}{\cos \theta}$$

Pulling everything to one side and factoring, we have:

$$\frac{\sin^2 \theta - \sin \theta}{\cos \theta} = 0$$ $$\frac{\sin \theta (\sin \theta - 1)}{\cos \theta} = 0$$

Clearly, either $\sin \theta = 0$ or $\sin \theta = 1$.

The first occurs when $\theta = \pi k$, where $k$ is any integer.

The second occurs when $\displaystyle{\theta = \frac{\pi}{2} + 2\pi k}$, where again, $k$ is any integer.

Thus, our solution set is: $$\{\theta \in \mathbb{R} \, | \, \theta = \pi k \textrm{ or } \theta = \frac{\pi}{2} + 2\pi k \textrm{ where } \theta \textrm{ is any integer} \}$$