Prove that:
[tex]\[
\frac{\sin \theta + \cos \theta + 1}{\sin \theta + \cos \theta - 1} - \frac{1 + \sin \theta - \cos \theta}{1 - \sin \theta + \cos \theta}
\][/tex]



Answer :

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]