Find the value of `sqrt((1-sintheta)/(1+sintheta))+sqrt((1+sintheta)/(1-sintheta))`.

Options

A. `-2sec theta`
B. `2sectheta`
C. `2cosectheta`
D. `2tantheta`

Answer

To find the value of \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} + \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \), we will follow these steps:

### Step 1: Rationalize the first square root
We start with the first term:
\[ \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \]
To rationalize it, we multiply the numerator and denominator by \( 1 - \sin \theta \):
\[ \sqrt{\frac{(1 - \sin \theta)^2}{(1 + \sin \theta)(1 - \sin \theta)}} \]
The denominator simplifies using the difference of squares:
\[ 1 - \sin^2 \theta = \cos^2 \theta \]
Thus, we have:
\[ \sqrt{\frac{(1 - \sin \theta)^2}{\cos^2 \theta}} = \frac{1 - \sin \theta}{\cos \theta} \]

### Step 2: Rationalize the second square root
Now we rationalize the second term:
\[ \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \]
Similarly, we multiply the numerator and denominator by \( 1 + \sin \theta \):
\[ \sqrt{\frac{(1 + \sin \theta)^2}{(1 - \sin \theta)(1 + \sin \theta)}} \]
Again, the denominator simplifies using the difference of squares:
\[ 1 - \sin^2 \theta = \cos^2 \theta \]
Thus, we have:
\[ \sqrt{\frac{(1 + \sin \theta)^2}{\cos^2 \theta}} = \frac{1 + \sin \theta}{\cos \theta} \]

### Step 3: Combine the two terms
Now we can combine both terms:
\[ \frac{1 - \sin \theta}{\cos \theta} + \frac{1 + \sin \theta}{\cos \theta} \]
Since they have a common denominator, we can combine them:
\[ \frac{(1 - \sin \theta) + (1 + \sin \theta)}{\cos \theta} = \frac{2}{\cos \theta} \]

### Step 4: Simplify the expression
Now, we can simplify this expression:
\[ \frac{2}{\cos \theta} = 2 \sec \theta \]

### Final Answer
Thus, the value of \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} + \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \) is:
\[ \boxed{2 \sec \theta} \]