Establish the identity. $$ \begin{array}{l}1+\frac{ an ^{2} heta}{\sec heta+1}=\sec heta \ ext { Simplify the numerator of the fraction from the left side by applying the appropriate Pythagorean Identity. } \ 1+\frac{\sec ^{2} heta-1}{\sec heta+1} ext { (Do not factor.) } \ ext { Factor the numerator of the fraction from the previous step. } \ 1+\frac{\square}{\sec heta+1}\end{array} $$

Question

Establish the identity. $$ \begin{array}{l}1+\frac{	an ^{2} 	heta}{\sec 	heta+1}=\sec 	heta \ 	ext { Simplify the numerator of the fraction from the left side by applying the appropriate Pythagorean Identity. } \ 1+\frac{\sec ^{2} 	heta-1}{\sec 	heta+1} 	ext { (Do not factor.) } \ 	ext { Factor the numerator of the fraction from the previous step. } \ 1+\frac{\square}{\sec 	heta+1}\end{array} $$

UpStudy AI · Accepted Answer

To establish the identity

$$
1+\frac{\tan ^{2} \theta}{\sec \theta+1}=\sec \theta,
$$

we will follow the steps outlined in the problem.

### Step 1: Simplify the numerator of the fraction using the Pythagorean Identity

We know from the Pythagorean identity that

$$
\tan^2 \theta + 1 = \sec^2 \theta.
$$

Thus, we can express $$\tan^2 \theta$$ as:

$$
\tan^2 \theta = \sec^2 \theta - 1.
$$

Substituting this into the left side of the identity gives us:

$$
1 + \frac{\tan^2 \theta}{\sec \theta + 1} = 1 + \frac{\sec^2 \theta - 1}{\sec \theta + 1}.
$$

### Step 2: Factor the numerator of the fraction

Now we need to simplify the fraction:

$$
1 + \frac{\sec^2 \theta - 1}{\sec \theta + 1}.
$$

The numerator $$\sec^2 \theta - 1$$ can be factored as:

$$
\sec^2 \theta - 1 = (\sec \theta - 1)(\sec \theta + 1).
$$

Now substituting this back into the expression gives us:

$$
1 + \frac{(\sec \theta - 1)(\sec \theta + 1)}{\sec \theta + 1}.
$$

### Step 3: Simplify the expression

Since $$\sec \theta + 1$$ is in both the numerator and the denominator, we can cancel it out (assuming $$\sec \theta + 1 \neq 0$$):

$$
1 + (\sec \theta - 1).
$$

This simplifies to:

$$
\sec \theta.
$$

### Conclusion

Thus, we have shown that:

$$
1 + \frac{\tan^2 \theta}{\sec \theta + 1} = \sec \theta,
$$

which establishes the identity.
$$ 1 + \frac{\sec^2 \theta - 1}{\sec \theta + 1} = \sec \theta $$

Establish the identity.

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