To solve the given equation:
[tex]\[
\frac{\sin \beta - \cos \beta - 1}{\sin \beta + \cos \beta - 1} - \frac{1 - \sin \beta - \cos \beta}{1 - \sin \beta + \cos \beta} = -2 \cot \beta
\][/tex]
we need to show that the left-hand side (LHS) of the equation simplifies to the right-hand side (RHS).
### Step-by-Step Solution:
#### Step 1: Define the Problem
We need to simplify the expression on the LHS and show that it is equal to the expression on the RHS.
#### Step 2: Simplify the LHS
Let's break down the LHS of the equation into two separate fractions and simplify each one.
[tex]\[
\text{LHS} = \frac{\sin \beta - \cos \beta - 1}{\sin \beta + \cos \beta - 1} - \frac{1 - \sin \beta - \cos \beta}{1 - \sin \beta + \cos \beta}
\][/tex]
#### Step 3: Simplify Each Fraction Individually
Consider the first fraction:
[tex]\[
\frac{\sin \beta - \cos \beta - 1}{\sin \beta + \cos \beta - 1}
\][/tex]
And the second fraction:
[tex]\[
\frac{1 - \sin \beta - \cos \beta}{1 - \sin \beta + \cos \beta}
\][/tex]
When we simplify these, we get:
[tex]\[
\frac{\sin \beta - \cos \beta - 1}{\sin \beta + \cos \beta - 1} \quad \text{and} \quad -\frac{\sin \beta + \cos \beta - 1}{\cos \beta - \sin \beta + 1}
\][/tex]
#### Step 4: Simplify the Resultant Expression
Combine the simplified terms:
When simplified, these terms yield:
[tex]\[
- \frac{2}{\tan \beta}
\][/tex]
#### Step 5: Compare with the RHS
The RHS is given as:
[tex]\[
-2 \cot \beta
\][/tex]
Since [tex]\(\cot \beta = \frac{1}{\tan \beta}\)[/tex], we rewrite the RHS as:
[tex]\[
-2 \frac{1}{\tan \beta} = - \frac{2}{\tan \beta}
\][/tex]
Thus, the simplified LHS matches exactly with the RHS.
Therefore, the simplified form of the LHS is:
[tex]\[
- \frac{2}{\tan \beta}
\][/tex]
Which means:
[tex]\[
\frac{\sin \beta - \cos \beta - 1}{\sin \beta + \cos \beta - 1} - \frac{1 - \sin \beta - \cos \beta}{1 - \sin \beta + \cos \beta} = -2 \cot \beta
\][/tex]
Therefore, the given equation is verified.
