Simplest form of `tan^(-1)((sqrt(1+cos x)+sqrt(1-cos x))/(sqrt(1+cos x)-sqrt(1-cos x))), pi lt x lt (3 pi)/(2)` is:

Options

A. `(pi)/(4)+(x)/(2)`
B. `(3 pi)/(2)-(x)/(2)`
C. `-(x)/(2)`
D. `pi -(x)/(2)`

Answer

To simplify the expression \( \tan^{-1}\left(\frac{\sqrt{1 + \cos x} + \sqrt{1 - \cos x}}{\sqrt{1 + \cos x} - \sqrt{1 - \cos x}}\right) \) for \( \pi < x < \frac{3\pi}{2} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \tan^{-1}\left(\frac{\sqrt{1 + \cos x} + \sqrt{1 - \cos x}}{\sqrt{1 + \cos x} - \sqrt{1 - \cos x}}\right) \]

### Step 2: Use trigonometric identities
We can use the identities:
\[ \sqrt{1 + \cos x} = \sqrt{2} \cos\left(\frac{x}{2}\right) \]
\[ \sqrt{1 - \cos x} = \sqrt{2} \sin\left(\frac{x}{2}\right) \]
Substituting these into the expression gives:
\[ y = \tan^{-1}\left(\frac{\sqrt{2} \cos\left(\frac{x}{2}\right) + \sqrt{2} \sin\left(\frac{x}{2}\right)}{\sqrt{2} \cos\left(\frac{x}{2}\right) - \sqrt{2} \sin\left(\frac{x}{2}\right)}\right) \]

### Step 3: Simplify the fraction
Factoring out \(\sqrt{2}\) from the numerator and denominator:
\[ y = \tan^{-1}\left(\frac{\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right)}\right) \]

### Step 4: Use the tangent addition formula
The expression can be interpreted using the tangent addition formula:
\[ \tan\left(\frac{\pi}{4} + \frac{x}{2}\right) = \frac{\tan\left(\frac{\pi}{4}\right) + \tan\left(\frac{x}{2}\right)}{1 - \tan\left(\frac{\pi}{4}\right) \tan\left(\frac{x}{2}\right)} \]
Since \(\tan\left(\frac{\pi}{4}\right) = 1\), we have:
\[ y = \tan^{-1}\left(\tan\left(\frac{\pi}{4} + \frac{x}{2}\right)\right) \]

### Step 5: Simplify the final expression
Thus, we can conclude:
\[ y = \frac{\pi}{4} + \frac{x}{2} \]

### Step 6: Consider the range of x
Given that \( \pi < x < \frac{3\pi}{2} \), we can find the range of \( y \):
- When \( x = \pi \), \( y = \frac{\pi}{4} + \frac{\pi}{2} = \frac{3\pi}{4} \)
- When \( x = \frac{3\pi}{2} \), \( y = \frac{\pi}{4} + \frac{3\pi}{4} = \pi \)

Thus, the simplest form of the expression is:
\[ \frac{\pi}{4} + \frac{x}{2}, \quad \text{for } \pi < x < \frac{3\pi}{2} \]