The value of `(sin A+sinB)/(cos A+cos B)+(cos A-cosB)/(sinA-sinB)` is……

Answer

To solve the expression \(\frac{\sin A + \sin B}{\cos A + \cos B} + \frac{\cos A - \cos B}{\sin A - \sin B}\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((\cos A + \cos B)(\sin A - \sin B)\).

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ \frac{(\sin A + \sin B)(\sin A - \sin B) + (\cos A - \cos B)(\cos A + \cos B)}{(\cos A + \cos B)(\sin A - \sin B)} \]

### Step 3: Simplify the numerator
Using the difference of squares formula, we can simplify the numerator:
\[ (\sin A + \sin B)(\sin A - \sin B) = \sin^2 A - \sin^2 B \]
\[ (\cos A - \cos B)(\cos A + \cos B) = \cos^2 A - \cos^2 B \]
Thus, the numerator becomes:
\[ \sin^2 A - \sin^2 B + \cos^2 A - \cos^2 B \]

### Step 4: Use the Pythagorean identity
We know that \(\sin^2 A + \cos^2 A = 1\) and \(\sin^2 B + \cos^2 B = 1\). Therefore:
\[ \sin^2 A - \sin^2 B + \cos^2 A - \cos^2 B = (1 - \sin^2 B) - (1 - \sin^2 A) = \sin^2 A - \sin^2 B \]
This simplifies to:
\[ (\sin^2 A - \sin^2 B) + (\cos^2 A - \cos^2 B) = 0 \]

### Step 5: Substitute back into the expression
Now substituting back, we have:
\[ \frac{0}{(\cos A + \cos B)(\sin A - \sin B)} = 0 \]

### Final Answer
Thus, the value of the given expression is:
\[ \boxed{0} \]