`(sintheta)/(cottheta+cosectheta)-(sintheta)/(cottheta-cosectheta)=?`

Options

A. 1
B. 2
C. 3
D. 0

Answer

To solve the expression

\[ \frac{\sin \theta}{\cot \theta + \csc \theta} - \frac{\sin \theta}{\cot \theta - \csc \theta} \]

we will follow these steps:

### Step 1: Write the expression clearly
We start with the expression:

\[ \frac{\sin \theta}{\cot \theta + \csc \theta} - \frac{\sin \theta}{\cot \theta - \csc \theta} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is:

\[ (\cot \theta + \csc \theta)(\cot \theta - \csc \theta) \]

### Step 3: Rewrite the expression with the common denominator
We can rewrite the expression as:

\[ \frac{\sin \theta (\cot \theta - \csc \theta) - \sin \theta (\cot \theta + \csc \theta)}{(\cot \theta + \csc \theta)(\cot \theta - \csc \theta)} \]

### Step 4: Simplify the numerator
Now, simplify the numerator:

\[ \sin \theta (\cot \theta - \csc \theta) - \sin \theta (\cot \theta + \csc \theta) = \sin \theta \cot \theta - \sin \theta \csc \theta - \sin \theta \cot \theta - \sin \theta \csc \theta \]

This simplifies to:

\[ -2 \sin \theta \csc \theta \]

### Step 5: Substitute \(\csc \theta\)
Recall that \(\csc \theta = \frac{1}{\sin \theta}\), so we can substitute:

\[ -2 \sin \theta \cdot \frac{1}{\sin \theta} = -2 \]

### Step 6: Simplify the denominator
The denominator can be simplified using the difference of squares:

\[ (\cot \theta + \csc \theta)(\cot \theta - \csc \theta) = \cot^2 \theta - \csc^2 \theta \]

### Step 7: Use the identity
Using the identity \(\csc^2 \theta = 1 + \cot^2 \theta\), we can rewrite the denominator:

\[ \cot^2 \theta - (1 + \cot^2 \theta) = -1 \]

### Step 8: Final expression
Now we have:

\[ \frac{-2}{-1} = 2 \]

Thus, the final answer is:

\[ \boxed{2} \]