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

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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)
1 + (sec² θ − 1) ⁄ (sec θ + 1)

Establish the identity.

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