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 value of \( \sin \theta \)
Since \( \theta \) is in the second quadrant, we can choose a specific angle for simplicity. Let's take \( \theta = 150^\circ \).

From trigonometric values, we know:
\[ \sin 150^\circ = \frac{1}{2} \]

### Step 2: Substitute \( \sin \theta \) into the expression
Now we substitute \( \sin \theta \) into the expression:
\[ \sqrt{\frac{1 - \frac{1}{2}}{1 + \frac{1}{2}}} + \sqrt{\frac{1 + \frac{1}{2}}{1 - \frac{1}{2}}} \]

### Step 3: Simplify the fractions
Calculating each part:
1. For the first term:
\[ \sqrt{\frac{1 - \frac{1}{2}}{1 + \frac{1}{2}}} = \sqrt{\frac{\frac{1}{2}}{\frac{3}{2}}} = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]

2. For the second term:
\[ \sqrt{\frac{1 + \frac{1}{2}}{1 - \frac{1}{2}}} = \sqrt{\frac{\frac{3}{2}}{\frac{1}{2}}} = \sqrt{3} \]

### Step 4: Combine the results
Now we combine the results from the two terms:
\[ \frac{1}{\sqrt{3}} + \sqrt{3} \]

### Step 5: Find a common denominator
To add these two fractions, we can express \( \sqrt{3} \) with a common denominator:
\[ \sqrt{3} = \frac{3}{\sqrt{3}} \]
Thus, we can write:
\[ \frac{1}{\sqrt{3}} + \frac{3}{\sqrt{3}} = \frac{1 + 3}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]

### Step 6: Final result
The final result is:
\[ \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} + \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} = \frac{4}{\sqrt{3}} \]