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

Options

A. `2sectheta`
B. `sectheta`
C. `2cosectheta`
D. None of these

Answer

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

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

### Step 2: Rationalize the first term
For the first term, we can rationalize the denominator:
\[ \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} = \frac{\sqrt{(1 + \cos \theta)^2}}{\sqrt{(1 - \cos \theta)(1 + \cos \theta)}} \]
Using the identity \( 1 - \cos^2 \theta = \sin^2 \theta \):
\[ = \frac{1 + \cos \theta}{\sqrt{\sin^2 \theta}} = \frac{1 + \cos \theta}{|\sin \theta|} \]

### Step 3: Rationalize the second term
Now we rationalize the second term:
\[ \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\sqrt{(1 - \cos \theta)^2}}{\sqrt{(1 + \cos \theta)(1 - \cos \theta)}} \]
Again using the identity \( 1 - \cos^2 \theta = \sin^2 \theta \):
\[ = \frac{1 - \cos \theta}{\sqrt{\sin^2 \theta}} = \frac{1 - \cos \theta}{|\sin \theta|} \]

### Step 4: Combine the two terms
Now we can combine the two terms:
\[ \frac{1 + \cos \theta}{|\sin \theta|} + \frac{1 - \cos \theta}{|\sin \theta|} = \frac{(1 + \cos \theta) + (1 - \cos \theta)}{|\sin \theta|} = \frac{2}{|\sin \theta|} \]

### Step 5: Simplify the expression
Thus, the expression simplifies to:
\[ \frac{2}{|\sin \theta|} = 2 \cdot \frac{1}{|\sin \theta|} = 2 \cdot \csc \theta \]

### Final Result
The final value of the original expression is:
\[ 2 \cdot \csc \theta \]