The simplified value of `1- (sin^(2)A)/(1+ cosA) + (1+ cosA)/(sinA) - (sinA)/(1- cosA)` is:

Options

A. 0
B. 1
C. sinA
D. cosA

Answer

To simplify the expression \(1 - \frac{\sin^2 A}{1 + \cos A} + \frac{1 + \cos A}{\sin A} - \frac{\sin A}{1 - \cos A}\), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Group the Terms**:
We can group the expression into two parts:
\[ \left(1 - \frac{\sin^2 A}{1 + \cos A}\right) + \left(\frac{1 + \cos A}{\sin A} - \frac{\sin A}{1 - \cos A}\right) \]

2. **Simplify the First Group**:
For the first part:
\[ 1 - \frac{\sin^2 A}{1 + \cos A} \]
We know that \(\sin^2 A = 1 - \cos^2 A\). Thus:
\[ 1 - \frac{1 - \cos^2 A}{1 + \cos A} = 1 - \left(\frac{1}{1 + \cos A} - \frac{\cos^2 A}{1 + \cos A}\right) \]
This simplifies to:
\[ 1 - \frac{1}{1 + \cos A} + \frac{\cos^2 A}{1 + \cos A} \]

3. **Combine the Terms**:
The common denominator for the first group is \(1 + \cos A\):
\[ \frac{(1 + \cos A) - 1 + \cos^2 A}{1 + \cos A} = \frac{\cos A + \cos^2 A}{1 + \cos A} = \frac{\cos A(1 + \cos A)}{1 + \cos A} = \cos A \]

4. **Simplify the Second Group**:
Now, for the second part:
\[ \frac{1 + \cos A}{\sin A} - \frac{\sin A}{1 - \cos A} \]
The common denominator here is \(\sin A(1 - \cos A)\):
\[ \frac{(1 + \cos A)(1 - \cos A) - \sin^2 A}{\sin A(1 - \cos A)} \]
Using the identity \(\sin^2 A = 1 - \cos^2 A\):
\[ (1 + \cos A)(1 - \cos A) - (1 - \cos^2 A) = 1 - \cos^2 A - 1 + \cos^2 A = 0 \]

5. **Combine Both Parts**:
Now combining both parts:
\[ \cos A + 0 = \cos A \]

### Final Answer:
Thus, the simplified value of the expression is:
\[ \cos A \]