If `theta` lies in the second quadrant. Then the value of `sqrt((1-sin theta)/(1+sin theta))+sqrt((1+sin theta)/(1-sin theta))` is equal to :

Options

A. `2 sec theta`
B. `-2 sec theta`
C. `2"cosec"theta`
D. 2

Answer

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

### Step 1: Identify the properties of sine and cosine in the second quadrant
In the second quadrant, \( \sin \theta \) is positive and \( \cos \theta \) is negative.

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} + \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \]

### Step 3: Combine the two terms under a common denominator
To combine the two square roots, we can express them with a common denominator:
\[ \sqrt{\frac{(1 - \sin \theta)(1 - \sin \theta) + (1 + \sin \theta)(1 + \sin \theta)}{(1 + \sin \theta)(1 - \sin \theta)}} \]

### Step 4: Simplify the numerator
Now, let's simplify the numerator:
\[ (1 - \sin \theta) + (1 + \sin \theta) = 1 - \sin \theta + 1 + \sin \theta = 2 \]

### Step 5: Simplify the denominator
The denominator can be simplified using the difference of squares:
\[ (1 + \sin \theta)(1 - \sin \theta) = 1 - \sin^2 \theta = \cos^2 \theta \]

### Step 6: Substitute back into the expression
Now, we can substitute back into the expression:
\[ \sqrt{\frac{2}{\cos^2 \theta}} = \frac{\sqrt{2}}{|\cos \theta|} \]

### Step 7: Determine the absolute value of cosine
Since \( \theta \) is in the second quadrant, \( \cos \theta \) is negative. Therefore, \( |\cos \theta| = -\cos \theta \).

### Step 8: Final expression
Thus, we have:
\[ \frac{\sqrt{2}}{-\cos \theta} = -\frac{\sqrt{2}}{\cos \theta} \]

### Conclusion
The final value of the expression is:
\[ -\frac{\sqrt{2}}{\cos \theta} \]