Find the value of `sqrt((1+cos theta)/(1-cos theta))+sqrt((1-cos theta)/(1+cos theta))`

Options

A. `2 sec theta`
B. `sec theta`
C. `"2 cosec "theta`
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 can follow these steps:

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

### Step 2: Combine the terms under a common denominator
To combine the two square root terms, we can express them with a common denominator:
\[ x = \frac{\sqrt{(1+\cos \theta)^2} + \sqrt{(1-\cos \theta)^2}}{\sqrt{(1-\cos \theta)(1+\cos \theta)}} \]

### Step 3: Simplify the numerator
The numerator simplifies to:
\[ \sqrt{(1+\cos \theta)^2} + \sqrt{(1-\cos \theta)^2} = (1+\cos \theta) + (1-\cos \theta) = 2 \]

### Step 4: Simplify the denominator
The denominator simplifies as follows:
\[ \sqrt{(1-\cos \theta)(1+\cos \theta)} = \sqrt{1 - \cos^2 \theta} = \sqrt{\sin^2 \theta} = |\sin \theta| \]
Since we are considering angles where \( \sin \theta \) is non-negative, we can write:
\[ \sqrt{\sin^2 \theta} = \sin \theta \]

### Step 5: Combine results
Now we can substitute back into our expression:
\[ x = \frac{2}{\sin \theta} \]

### Step 6: Final expression
Thus, we can express \( x \) as:
\[ x = 2 \cdot \csc \theta \]

### Conclusion
The final value of the expression is:
\[ \sqrt{\frac{1+\cos \theta}{1-\cos \theta}} + \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} = 2 \cdot \csc \theta \]