Answer
$$\frac{\sin^2\theta}{\cos\theta}=\sec\theta-\cos\theta$$
The trigonometric expression is an identity. We can prove by representing the right side in terms of $\sin\theta$ and $\cos\theta$.
Work Step by Step
$$\frac{\sin^2\theta}{\cos\theta}=\sec\theta-\cos\theta$$
We find that the left side comprises of only $\sin\theta$ and $\cos\theta$, while the right side includes $\sec\theta$. So the right side has more potential to be simplified, so we would simplify the right side first.
$$A=\sec\theta-\cos\theta$$
- Reciprocal Identity:
$$\sec\theta=\frac{1}{\cos\theta}$$
Apply it into $A$:
$$A=\frac{1}{\cos\theta}-\cos\theta$$
$$A=\frac{1-\cos^2\theta}{\cos\theta}$$
- Now we can use the Pythagorean Identity:
$$1-\cos^2\theta=\sin^2\theta$$
Therefore, $$A=\frac{\sin^2\theta}{\cos\theta}$$
The left side and right side are thus equal. The given trigonometric expression is an identity.
