To prove the equality:
[tex]\[
\frac{\sin \theta + \cos \theta + 1}{\sin \theta + \cos \theta - 1} = \frac{1 + \sin \theta - \cos \theta}{1 - \sin \theta + \cos \theta}
\][/tex]
we start by simplifying each side separately.
First, consider the left-hand side of the expression:
[tex]\[
\frac{\sin \theta + \cos \theta + 1}{\sin \theta + \cos \theta - 1}
\][/tex]
To simplify this, recognize that:
[tex]\[
\sin \theta + \cos \theta = \sqrt{2} \sin \left( \theta + \frac{\pi}{4} \right)
\][/tex]
Taking this into consideration, the expression becomes:
[tex]\[
\frac{\sqrt{2} \sin \left( \theta + \frac{\pi}{4} \right) + 1}{\sqrt{2} \sin \left( \theta + \frac{\pi}{4} \right) - 1}
\][/tex]
Next, consider the right-hand side of the expression:
[tex]\[
\frac{1 + \sin \theta - \cos \theta}{1 - \sin \theta + \cos \theta}
\][/tex]
Here, note that:
[tex]\[
1 + \sin \theta - \cos \theta = 1 - \cos \theta + \sin \theta
\][/tex]
and similarly,
[tex]\[
1 - \sin \theta + \cos \theta = 1 - (\sin \theta - \cos \theta)
\][/tex]
Transforming in terms of [tex]\( \theta + \frac{\pi}{4} \)[/tex]:
[tex]\[
1 + \sin \theta - \cos \theta = -\sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) + 1
\][/tex]
and
[tex]\[
1 - \sin \theta + \cos \theta = \sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) + 1
\][/tex]
Therefore, the expression becomes:
[tex]\[
\frac{-\sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) + 1}{\sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) + 1}
\][/tex]
Now, comparing both sides:
[tex]\[
\frac{\sqrt{2} \sin \left( \theta + \frac{\pi}{4} \right) + 1}{\sqrt{2} \sin \left( \theta + \frac{\pi}{4} \right) - 1}
\][/tex]
[tex]\[
\frac{-\sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) + 1}{\sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) + 1}
\][/tex]
We see that these two expressions are simplified forms of the original ones given in the problem. They reduce to
[tex]\[
\frac{\sin \theta + \cos \theta + 1}{\sin \theta + \cos \theta - 1} = \frac{-\sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) + 1}{\sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) + 1}
\][/tex]
Therefore, we have shown that:
[tex]\[
\boxed{\frac{\sin \theta + \cos \theta + 1}{\sin \theta + \cos \theta - 1} = \frac{1 + \sin \theta - \cos \theta}{1 - \sin \theta + \cos \theta}}
\][/tex]
