If `theta` lies in the second quadrant, then `sqrt((1-sintheta)/(1+sin theta))+sqrt((1+sin theta)/(1-sin theta))` is equal to :

Options

A. `-2sec theta`
B. `1//2`
C. `sqrt3`
D. `2 tan theta`

Answer

To solve the expression \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} + \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \) where \( \theta \) lies in the second quadrant, we can follow these steps:

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

### Step 2: Combine the fractions
We can combine the two square roots into a single fraction:
\[ \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} + \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} = \frac{\sqrt{(1 - \sin \theta)(1 - \sin \theta)} + \sqrt{(1 + \sin \theta)(1 + \sin \theta)}}{\sqrt{(1 + \sin \theta)(1 - \sin \theta)}} \]
This simplifies to:
\[ \frac{\sqrt{(1 - \sin \theta)^2} + \sqrt{(1 + \sin \theta)^2}}{\sqrt{(1 + \sin \theta)(1 - \sin \theta)}} \]

### Step 3: Simplify the numerator
The numerator becomes:
\[ |1 - \sin \theta| + |1 + \sin \theta| \]
Since \( \theta \) is in the second quadrant, \( \sin \theta \) is positive. Therefore:
\[ |1 - \sin \theta| = 1 - \sin \theta \quad \text{and} \quad |1 + \sin \theta| = 1 + \sin \theta \]
Thus, the numerator simplifies to:
\[ (1 - \sin \theta) + (1 + \sin \theta) = 2 \]

### Step 4: Simplify the denominator
The denominator becomes:
\[ \sqrt{(1 + \sin \theta)(1 - \sin \theta)} = \sqrt{1 - \sin^2 \theta} = \sqrt{\cos^2 \theta} = |\cos \theta| \]
Since \( \theta \) is in the second quadrant, \( \cos \theta \) is negative, so:
\[ |\cos \theta| = -\cos \theta \]

### Step 5: Final expression
Now we can substitute back into our expression:
\[ \frac{2}{-\cos \theta} \]
This simplifies to:
\[ -\frac{2}{\cos \theta} \]

### Step 6: Conclusion
Thus, the final value of the expression is:
\[ -\frac{2}{\cos \theta} \]