`m cos ( theta + alpha ) = n cos ( theta - alpha ) `, then `( m + n )/( m -n ) = ( cot theta )/ ( cot alpha )`

Answer

To solve the equation \( m \cos(\theta + \alpha) = n \cos(\theta - \alpha) \) and prove that \( \frac{m+n}{m-n} = \frac{\cot \theta}{\cot \alpha} \), we can follow these steps:

### Step 1: Rewrite the Given Equation
We start with the given equation:
\[ m \cos(\theta + \alpha) = n \cos(\theta - \alpha) \]

### Step 2: Use the Cosine Addition and Subtraction Formulas
Using the cosine addition and subtraction formulas, we can expand both sides:
\[ m (\cos \theta \cos \alpha - \sin \theta \sin \alpha) = n (\cos \theta \cos \alpha + \sin \theta \sin \alpha) \]

### Step 3: Rearranging the Equation
Rearranging gives us:
\[ m \cos \theta \cos \alpha - m \sin \theta \sin \alpha = n \cos \theta \cos \alpha + n \sin \theta \sin \alpha \]
Now, we can group the terms involving \( \cos \theta \) and \( \sin \theta \):
\[ m \cos \theta \cos \alpha - n \cos \theta \cos \alpha = m \sin \theta \sin \alpha + n \sin \theta \sin \alpha \]
This simplifies to:
\[ (m - n) \cos \theta \cos \alpha = (m + n) \sin \theta \sin \alpha \]

### Step 4: Divide Both Sides
Now, we divide both sides by \( (m - n) \sin \alpha \) (assuming \( m \neq n \)):
\[ \frac{(m + n) \sin \theta}{(m - n) \sin \alpha} = \cos \theta \cos \alpha \]

### Step 5: Rearranging to Find the Desired Ratio
Rearranging gives us:
\[ \frac{m + n}{m - n} = \frac{\sin \theta \cos \alpha}{\sin \alpha \cos \theta} \]

### Step 6: Use the Definition of Cotangent
Using the definitions of cotangent, we can rewrite the right-hand side:
\[ \frac{m + n}{m - n} = \frac{\cot \theta}{\cot \alpha} \]

### Conclusion
Thus, we have shown that:
\[ \frac{m + n}{m - n} = \frac{\cot \theta}{\cot \alpha} \]