If sin A = `(2mn)/(m ^2+n^2)`, then find the value of `(sinAcotA)/(cosA)`

Answer

To solve the problem, we need to find the value of \(\frac{\sin A \cdot \cot A}{\cos A}\) given that \(\sin A = \frac{2mn}{m^2 + n^2}\).

### Step 1: Understand the cotangent function
The cotangent function is defined as:
\[ \cot A = \frac{\cos A}{\sin A} \]

### Step 2: Substitute cotangent in the expression
Now, we can substitute \(\cot A\) into the expression:
\[ \frac{\sin A \cdot \cot A}{\cos A} = \frac{\sin A \cdot \frac{\cos A}{\sin A}}{\cos A} \]

### Step 3: Simplify the expression
Notice that \(\sin A\) in the numerator and denominator cancels out:
\[ = \frac{\cos A}{\cos A} \]

### Step 4: Final simplification
Since \(\frac{\cos A}{\cos A} = 1\) (as long as \(\cos A \neq 0\)):
\[ \frac{\sin A \cdot \cot A}{\cos A} = 1 \]

### Final Answer
Thus, the value of \(\frac{\sin A \cdot \cot A}{\cos A}\) is:
\[ \boxed{1} \]