Eliminate `theta` from the equations
`tan(n theta+alpha)-tan(n theta+beta)=x` and
`cot(n theta+alpha)-cot(n theta+beta)=y`.

Answer

To eliminate \( \theta \) from the given equations:

1. **Start with the given equations:**
\[ \tan(n\theta + \alpha) - \tan(n\theta + \beta) = x \]
\[ \cot(n\theta + \alpha) - \cot(n\theta + \beta) = y \]

2. **Rewrite the second equation using the cotangent identity:**
Recall that \( \cot A = \frac{1}{\tan A} \). Thus, we can rewrite the second equation as:
\[ \frac{1}{\tan(n\theta + \alpha)} - \frac{1}{\tan(n\theta + \beta)} = y \]

3. **Combine the terms in the second equation:**
Taking the common denominator:
\[ \frac{\tan(n\theta + \beta) - \tan(n\theta + \alpha)}{\tan(n\theta + \alpha) \tan(n\theta + \beta)} = y \]
This can be rearranged to:
\[ \tan(n\theta + \beta) - \tan(n\theta + \alpha) = y \tan(n\theta + \alpha) \tan(n\theta + \beta) \]

4. **Substituting the first equation into the second:**
From the first equation, we have:
\[ \tan(n\theta + \beta) - \tan(n\theta + \alpha) = x \]
Substitute this into the rearranged second equation:
\[ x = y \tan(n\theta + \alpha) \tan(n\theta + \beta) \]

5. **Express \( \tan(n\theta + \alpha) \tan(n\theta + \beta) \) in terms of \( x \) and \( y \):**
Rearranging gives:
\[ \tan(n\theta + \alpha) \tan(n\theta + \beta) = \frac{x}{y} \]

6. **Use the tangent subtraction formula:**
Recall the formula for the tangent of a difference:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Here, let \( A = n\theta + \alpha \) and \( B = n\theta + \beta \):
\[ \tan(n\theta + \alpha - (n\theta + \beta)) = \frac{\tan(n\theta + \alpha) - \tan(n\theta + \beta)}{1 + \tan(n\theta + \alpha) \tan(n\theta + \beta)} \]

7. **Substituting the known values:**
This simplifies to:
\[ \tan(\alpha - \beta) = \frac{x}{1 + \frac{x}{y}} = \frac{xy}{y + x} \]

8. **Final equation without \( \theta \):**
Thus, we arrive at the final equation:
\[ \tan(\alpha - \beta) = \frac{xy}{y - x} \]