Answer
$$\frac{\tan^2\alpha+1}{\sec\alpha}=\sec\alpha$$
By using Pythagorean Identity $$\tan^2\alpha+1=\sec^2\alpha$$ for the numerator of the left side, it is proved to be an identity.
Work Step by Step
$$\frac{\tan^2\alpha+1}{\sec\alpha}=\sec\alpha$$
The left side is more complex, so we would simplify it first.
$$A=\frac{\tan^2\alpha+1}{\sec\alpha}$$
According to a Pythagorean Identity,
$$\tan^2\alpha+1=\sec^2\alpha$$
$A$ would become
$$A=\frac{\sec^2\alpha}{\sec\alpha}$$
$$A=\sec\alpha$$
The left side has been simplified to the right side. It is thus an identity.