Answer
$\frac{\pi}{4}+k\pi$ and $\frac{3\pi}{4}+k\pi$
Work Step by Step
Solve the equations by factoring
Step 1. Use the reciprocal relations $sec\theta=\frac{1}{cos\theta}$ and $tan\theta=\frac{sin\theta}{cos\theta}, cot\theta=\frac{cos\theta}{sin\theta}$,
we can rewrite the equation as: $\frac{sin\theta}{cos^2\theta}-\frac{cos^2\theta}{sin\theta}=sin\theta$
Step 2. Multiple $sin\theta cos^2\theta$ on both sides of the equation to obtain:
$sin^2\theta-cos^4\theta=sin^2\theta cos^2\theta$
Step 3. Rearrange the terms as $sin^2\theta-sin^2\theta cos^2\theta=cos^4\theta$ and factor the left side to get
$sin^2\theta(1-cos^2\theta)=cos^4\theta$. Use the Pythagorean Identity $sin^2\theta=1-cos^2\theta$, the equation becomes: $sin^4\theta=cos^4\theta$ or $tan^4\theta=1$
Step 4. Solve the above equation to obtain $tan\theta=\pm1$ and the general form for all the solutions are $\theta=\frac{\pi}{4}+k\pi$ and $\theta=\frac{3\pi}{4}+k\pi$ where $k$ is any integer.
