Find the value of `(1+cos A)(1-cos A)cosec^(2)A`.

Answer

To solve the expression \((1 + \cos A)(1 - \cos A) \csc^2 A\), we can follow these steps:

### Step 1: Use the difference of squares formula
We recognize that \((1 + \cos A)(1 - \cos A)\) is in the form of \( (a + b)(a - b) = a^2 - b^2 \). Here, \(a = 1\) and \(b = \cos A\).

\[ (1 + \cos A)(1 - \cos A) = 1^2 - (\cos A)^2 = 1 - \cos^2 A \]

### Step 2: Apply the Pythagorean identity
We know from the Pythagorean identity that:

\[ \sin^2 A + \cos^2 A = 1 \]

Thus, we can express \(1 - \cos^2 A\) as:

\[ 1 - \cos^2 A = \sin^2 A \]

### Step 3: Substitute back into the expression
Now we substitute \(\sin^2 A\) back into the original expression:

\[ (1 + \cos A)(1 - \cos A) \csc^2 A = \sin^2 A \csc^2 A \]

### Step 4: Simplify using the definition of cosecant
Recall that \(\csc A = \frac{1}{\sin A}\), therefore:

\[ \csc^2 A = \frac{1}{\sin^2 A} \]

Substituting this into our expression gives:

\[ \sin^2 A \cdot \csc^2 A = \sin^2 A \cdot \frac{1}{\sin^2 A} \]

### Step 5: Final simplification
The \(\sin^2 A\) in the numerator and denominator cancels out:

\[ \sin^2 A \cdot \frac{1}{\sin^2 A} = 1 \]

### Conclusion
Thus, the value of \((1 + \cos A)(1 - \cos A) \csc^2 A\) is:

\[ \boxed{1} \]