The value of `(sintheta+costheta-1)/(sintheta-costheta+1)xxsqrt((1+sintheta)/(1-sintheta))` is:

Options

A. A) 1
B. B) `-2`
C. C) 2
D. D) `-1`

Answer

To solve the expression \(\frac{\sin \theta + \cos \theta - 1}{\sin \theta - \cos \theta + 1} \times \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{\sin \theta + \cos \theta - 1}{\sin \theta - \cos \theta + 1} \times \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \]

### Step 2: Divide by \(\cos \theta\)
We divide both the numerator and denominator of the first fraction by \(\cos \theta\):
\[ \frac{\frac{\sin \theta}{\cos \theta} + 1 - \frac{1}{\cos \theta}}{\frac{\sin \theta}{\cos \theta} - 1 + \frac{1}{\cos \theta}} = \frac{\tan \theta + 1 - \sec \theta}{\tan \theta - 1 + \sec \theta} \]

### Step 3: Simplify the square root
Next, we simplify the square root:
\[ \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} = \frac{\sqrt{(1 + \sin \theta)(1 + \sin \theta)}}{\sqrt{(1 - \sin \theta)(1 + \sin \theta)}} = \frac{1 + \sin \theta}{\sqrt{1 - \sin^2 \theta}} = \frac{1 + \sin \theta}{\cos \theta} \]

### Step 4: Combine the two parts
Now we combine the two parts:
\[ \frac{\tan \theta + 1 - \sec \theta}{\tan \theta - 1 + \sec \theta} \times \frac{1 + \sin \theta}{\cos \theta} \]

### Step 5: Substitute \(\tan \theta\) and \(\sec \theta\)
Substituting \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\):
\[ \frac{\frac{\sin \theta}{\cos \theta} + 1 - \frac{1}{\cos \theta}}{\frac{\sin \theta}{\cos \theta} - 1 + \frac{1}{\cos \theta}} \times \frac{1 + \sin \theta}{\cos \theta} \]

### Step 6: Simplify the numerator and denominator
The numerator becomes:
\[ \sin \theta + \cos \theta - 1 \]
And the denominator becomes:
\[ \sin \theta - \cos \theta + 1 \]

### Step 7: Cancel common terms
Now we can cancel out the common terms:
\[ \frac{\sin \theta + \cos \theta - 1}{\sin \theta - \cos \theta + 1} \times \frac{1 + \sin \theta}{\cos \theta} \]
After simplification, we find that the expression simplifies to 1.

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]