`((1-sintheta+costheta)^(2))/((1+costheta)(1-sintheta))=?`

Options

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

Answer

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

### Step 1: Expand the numerator
The numerator is \((1 - \sin \theta + \cos \theta)^2\). We can expand this using the formula \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc)\).

Let:
- \(a = 1\)
- \(b = -\sin \theta\)
- \(c = \cos \theta\)

So,
\[ (1 - \sin \theta + \cos \theta)^2 = 1^2 + (-\sin \theta)^2 + (\cos \theta)^2 + 2(1 \cdot (-\sin \theta) + 1 \cdot \cos \theta + (-\sin \theta) \cdot \cos \theta) \]
Calculating each term:
- \(1^2 = 1\)
- \((- \sin \theta)^2 = \sin^2 \theta\)
- \((\cos \theta)^2 = \cos^2 \theta\)

Thus,
\[ = 1 + \sin^2 \theta + \cos^2 \theta + 2(-\sin \theta + \cos \theta - \sin \theta \cos \theta) \]
Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ = 1 + 1 + 2(-\sin \theta + \cos \theta - \sin \theta \cos \theta) \]
\[ = 2 + 2(-\sin \theta + \cos \theta - \sin \theta \cos \theta) \]

### Step 2: Simplify the denominator
The denominator is \((1 + \cos \theta)(1 - \sin \theta)\). We can expand this as follows:
\[ (1 + \cos \theta)(1 - \sin \theta) = 1 - \sin \theta + \cos \theta - \sin \theta \cos \theta \]

### Step 3: Substitute back into the expression
Now, we substitute the expanded numerator and denominator back into the expression:
\[ \frac{2 + 2(-\sin \theta + \cos \theta - \sin \theta \cos \theta)}{1 - \sin \theta + \cos \theta - \sin \theta \cos \theta} \]

### Step 4: Factor out common terms
Notice that both the numerator and denominator have common terms. We can factor out \(2\) from the numerator:
\[ = \frac{2(1 - \sin \theta + \cos \theta - \sin \theta \cos \theta)}{1 - \sin \theta + \cos \theta - \sin \theta \cos \theta} \]

### Step 5: Cancel common factors
Since the numerator and denominator are the same (except for the factor of 2), we can cancel them out:
\[ = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]