`1-(sin^(2)y)/(1+cosy)+(1+cosy)/(siny)-(siny)/(1-cosy)=?`

Options

A. 0
B. 1
C. sin y
D. cos y

Answer

To solve the expression \( 1 - \frac{\sin^2 y}{1 + \cos y} + \frac{1 + \cos y}{\sin y} - \frac{\sin y}{1 - \cos y} \), we will simplify it step by step.

### Step 1: Write down the expression
We start with the expression:
\[ 1 - \frac{\sin^2 y}{1 + \cos y} + \frac{1 + \cos y}{\sin y} - \frac{\sin y}{1 - \cos y} \]

### Step 2: Find a common denominator for the fractions
The common denominator for the fractions \( \frac{\sin^2 y}{1 + \cos y} \), \( \frac{1 + \cos y}{\sin y} \), and \( \frac{\sin y}{1 - \cos y} \) can be taken as \( \sin y (1 + \cos y)(1 - \cos y) \).

### Step 3: Rewrite each term with the common denominator
Rewriting the terms:
1. The first term becomes:
\[ 1 \cdot \frac{\sin y (1 + \cos y)(1 - \cos y)}{\sin y (1 + \cos y)(1 - \cos y)} = \frac{\sin y (1 + \cos y)(1 - \cos y)}{\sin y (1 + \cos y)(1 - \cos y)} \]
2. The second term becomes:
\[ -\frac{\sin^2 y (1 - \cos y)}{\sin y (1 + \cos y)(1 - \cos y)} \]
3. The third term becomes:
\[ \frac{(1 + \cos y)(1 - \cos y)}{(1 + \cos y)(1 - \cos y)} = \frac{1 - \cos^2 y}{\sin y (1 + \cos y)(1 - \cos y)} = \frac{\sin^2 y}{\sin y (1 + \cos y)(1 - \cos y)} \]
4. The fourth term becomes:
\[ -\frac{\sin^2 y}{\sin y (1 + \cos y)(1 - \cos y)} \]

### Step 4: Combine the fractions
Now we combine all the terms:
\[ \frac{\sin y (1 + \cos y)(1 - \cos y) - \sin^2 y (1 - \cos y) + (1 - \cos^2 y) - \sin^2 y}{\sin y (1 + \cos y)(1 - \cos y)} \]

### Step 5: Simplify the numerator
The numerator simplifies to:
\[ \sin y (1 + \cos y)(1 - \cos y) - \sin^2 y (1 - \cos y) + \sin^2 y - \sin^2 y \]
This can be simplified further using the identity \( 1 - \cos^2 y = \sin^2 y \):
\[ \sin y (1 + \cos y)(1 - \cos y) = \sin y \sin^2 y \]

### Step 6: Factor out common terms
Now we have:
\[ \frac{\sin^2 y (1 + \cos y)}{\sin y (1 + \cos y)(1 - \cos y)} \]
Canceling \( \sin y \) and \( (1 + \cos y) \):
\[ \frac{\sin y}{1 - \cos y} \]

### Step 7: Final simplification
The final answer is:
\[ \cos y \]

Thus, the simplified expression is:
\[ \cos y \]