Question

verify that the equation is an identity
\\( \frac { 1 + \tan ^ { 2 } \alpha } { \sec \alpha } = \sec \alpha \\)
to verify the identity, start with the more complicated side and transform it to look like the other side. choose the correct transformations and transform the expression at each step
\\( \frac { 1 + \tan ^ { 2 } \alpha } { \sec \alpha } = \frac { \square } { \sec \alpha } \\)
apply a pythagorean identity
apply a quotient identity
apply an even - odd identity

Explanation:

Step1: Apply a Pythagorean identity

We know the Pythagorean identity \(1+\tan^{2}\alpha=\sec^{2}\alpha\). So, \(\frac{1 + \tan^{2}\alpha}{\sec\alpha}=\frac{\sec^{2}\alpha}{\sec\alpha}\)

Step2: Simplify the fraction

Using the rule of exponents \(\frac{a^{m}}{a^{n}}=a^{m - n}\) (here \(a = \sec\alpha\), \(m=2\), \(n = 1\)), \(\frac{\sec^{2}\alpha}{\sec\alpha}=\sec^{2-1}\alpha=\sec\alpha\)

Answer:

The left - hand side \(\frac{1+\tan^{2}\alpha}{\sec\alpha}\) simplifies to \(\sec\alpha\), which is equal to the right - hand side. So the equation \(\frac{1+\tan^{2}\alpha}{\sec\alpha}=\sec\alpha\) is an identity.