Establish the identity. \[ \begin{array}{l}1+\frac{\tan ^{2} \theta}{\sec \theta+1}=\sec \theta \\ \text { Simplify the numerator of the fraction from the left side by applying the appropriate Pythagorean Identity. } \\ 1+\frac{\square}{\sec \theta+1} \text { (Do not factor.) }\end{array} \]

To establish the identity: \[ 1 + \frac{\tan^2 \theta}{\sec \theta + 1} = \sec \theta \] we can simplify the numerator of the fraction on the left side using the appropriate Pythagorean identity. **Step 1: Apply the Pythagorean Identity** The relevant Pythagorean identity involving \(\tan^2 \theta\) is: \[ \tan^2 \theta + 1 = \sec^2 \theta \] Solving for \(\tan^2 \theta\), we get: \[ \tan^2 \theta = \sec^2 \theta - 1 \] **Step 2: Substitute Back into the Original Expression** Replace \(\tan^2 \theta\) in the original expression with \(\sec^2 \theta - 1\): \[ 1 + \frac{\sec^2 \theta - 1}{\sec \theta + 1} \] So, the identity after simplifying the numerator is: \[ 1 + \frac{\sec^2 \theta - 1}{\sec \theta + 1} \] **Final Answer:** After simplifying, the numerator is sec²θ minus one. Thus, 1 + (sec² θ − 1) ⁄ (sec θ + 1)