Answer
$$\frac{\tan\alpha}{\sec\alpha}=\sin\alpha$$
The trigonometric equation is proved to be an identity by simplifying the left side, using the Quotient and Reciprocal Identities.
Work Step by Step
$$\frac{\tan\alpha}{\sec\alpha}=\sin\alpha$$
We would start from the left side, since it is more complicated.
$$A=\frac{\tan\alpha}{\sec\alpha}$$
- Quotient Identity: $$\tan\alpha=\frac{\sin\alpha}{\cos\alpha}$$
- Reciprocal Identity:
$$\sec\alpha=\frac{1}{\cos\alpha}$$
Apply them into $A$, we have
$$A=\frac{\frac{\sin\alpha}{\cos\alpha}}{\frac{1}{\cos\alpha}}$$
$$A=\frac{\sin\alpha}{\cos\alpha}\times\cos\alpha$$
$$A=\sin\alpha$$
The left side is hence proved to be equal with the right side. The trigonometric equation is an identity.