Sure! Let's solve the given equation step by step: [tex]\(\cot \beta (\cot \alpha - 1) - \cot \alpha = 1\)[/tex].
### Step 1: Understand the Equation
The equation given to us is:
[tex]\[ \cot \beta (\cot \alpha - 1) - \cot \alpha = 1 \][/tex]
### Step 2: Isolate [tex]\(\cot \beta\)[/tex]
First, let's rewrite the equation to isolate the term involving [tex]\(\cot \beta\)[/tex]:
[tex]\[ \cot \beta (\cot \alpha - 1) - \cot \alpha = 1 \][/tex]
[tex]\[ \cot \beta (\cot \alpha - 1) = \cot \alpha + 1 \][/tex]
### Step 3: Solve for [tex]\(\cot \beta\)[/tex]
Next, we solve for [tex]\(\cot \beta\)[/tex] by dividing both sides of the equation by [tex]\((\cot \alpha - 1)\)[/tex]:
[tex]\[ \cot \beta = \frac{\cot \alpha + 1}{\cot \alpha - 1} \][/tex]
### Step 4: Find [tex]\(\beta\)[/tex] in terms of [tex]\(\alpha\)[/tex]
To find [tex]\(\beta\)[/tex] in terms of [tex]\(\alpha\)[/tex], we need to take the inverse cotangent ([tex]\(\text{acot}\)[/tex]) of both sides:
[tex]\[ \beta = \text{acot}\left( \frac{\cot \alpha + 1}{\cot \alpha - 1} \right) \][/tex]
Therefore, the solution for [tex]\(\beta\)[/tex] in terms of [tex]\(\alpha\)[/tex] is:
[tex]\[ \beta = \text{acot}\left( \frac{\cot \alpha + 1}{\cot \alpha - 1} \right) \][/tex]
This gives us the relationship between [tex]\(\beta\)[/tex] and [tex]\(\alpha\)[/tex] for the given equation.
