\frac{{1 + {\text {sin}}2\theta  + {\text {cos}}2\theta }}{{1 + {\text {sin}}2\theta  - {\text {cos}}2\theta }} is equals to

\frac{1}{2} tan θ

\frac{1}{2} cot θ

tan θ

cot θ

We are tasked with simplifying the expression:

\[
\frac{1 + \sin 2\theta + \cos 2\theta}{1 + \sin 2\theta - \cos 2\theta}
\]

and determining if it is equal to \(\frac{1}{2} \tan \theta\), \(\frac{1}{2} \cot \theta\), \(\tan \theta\), or \(\cot \theta\).

### Step 1: Rewrite using trigonometric identities
First, we recognize that \(\sin 2\theta\) and \(\cos 2\theta\) are the double-angle identities:
- \(\sin 2\theta = 2 \sin \theta \cos \theta\)
- \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)

However, to avoid complicating the process too much, let's approach the problem in a different way by using the sum formulas instead.

### Step 2: Simplifying the numerator and denominator
Use the identity:
\[
\cos 2\theta = 1 - 2\sin^2 \theta
\]
Now we can try to rewrite and reduce the original expression to match one of the answers (such as \(\tan \theta\)).

This can be done

The query submitted by the user: \frac{{1 + {\text {sin}}2\theta  + {\text {cos}}2\theta }}{{1 + {\text {sin}}2\theta  - {\text {cos}}2\theta }} is equals to \frac{1}{2} tan \theta, \frac{1}{2} cot \theta, tan \theta, or cot \theta?

Learn how to simplify the trigonometric expression \frac{1 + \sin 2\theta + \cos 2\theta}{1 + \sin 2\theta - \cos 2\theta} using double-angle identities. This problem explains key trigonometric concepts like \sin 2\theta and \cos 2\theta and helps you identify the correct simplified form.

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