If `(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))=cot(a+x/2)` and `x in (pi,2pi)` then 'a' is equal to :

Options

A. `(pi)/4`
B. `(pi)/2`
C. `(pi)/3`
D. None of these

Answer

To solve the equation

\[ \frac{\sqrt{1+\cos x}+\sqrt{1-\cos x}}{\sqrt{1+\cos x}-\sqrt{1-\cos x}}=\cot(a+\frac{x}{2}), \]

where \( x \) is in the interval \( ( \pi, 2\pi ) \), we will follow these steps:

### Step 1: Simplify the left-hand side
We start by rewriting \( \sqrt{1+\cos x} \) and \( \sqrt{1-\cos x} \) using trigonometric identities.

Using the half-angle identities:
\[ \sqrt{1+\cos x} = \sqrt{2\cos^2\left(\frac{x}{2}\right)} = \sqrt{2} \cos\left(\frac{x}{2}\right), \]
\[ \sqrt{1-\cos x} = \sqrt{2\sin^2\left(\frac{x}{2}\right)} = \sqrt{2} \sin\left(\frac{x}{2}\right). \]

### Step 2: Substitute into the equation
Now we substitute these into the left-hand side:
\[ \frac{\sqrt{2} \cos\left(\frac{x}{2}\right) + \sqrt{2} \sin\left(\frac{x}{2}\right)}{\sqrt{2} \cos\left(\frac{x}{2}\right) - \sqrt{2} \sin\left(\frac{x}{2}\right)}. \]

### Step 3: Factor out \(\sqrt{2}\)
Factoring out \(\sqrt{2}\) from both the numerator and the denominator:
\[ = \frac{\sqrt{2} \left( \cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right) \right)}{\sqrt{2} \left( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right) \right)} = \frac{\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right)}. \]

### Step 4: Rewrite in terms of cotangent
Now, we can rewrite this as:
\[ = \frac{1 + \tan\left(\frac{x}{2}\right)}{1 - \tan\left(\frac{x}{2}\right)}. \]

Using the identity for the tangent of a sum:
\[ \tan\left(\frac{\pi}{4} + \frac{x}{2}\right) = \frac{1 + \tan\left(\frac{x}{2}\right)}{1 - \tan\left(\frac{x}{2}\right)}. \]

### Step 5: Set the equation
Now we have:
\[ \cot\left(\frac{\pi}{4} + \frac{x}{2}\right) = \cot(a + \frac{x}{2}). \]

### Step 6: Compare angles
From the equation above, we can compare the angles:
\[ a + \frac{x}{2} = \frac{\pi}{4} + \frac{x}{2}. \]

### Step 7: Solve for \( a \)
Subtracting \(\frac{x}{2}\) from both sides gives:
\[ a = \frac{\pi}{4}. \]

Thus, the value of \( a \) is:
\[ \boxed{\frac{\pi}{4}}. \]