The value of `tan^(-1)[(sqrt(1-sinx)+sqrt(1+sinx))/(sqrt(1-sinx)-sqrt(1+sinx))](AA x in [0, (pi)/(2)])` is equal to

Options

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

Answer

To solve the problem, we need to evaluate the expression:

\[ y = \tan^{-1}\left(\frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}}\right) \]

**Step 1: Simplifying the expression inside the arctangent**

We can start by simplifying the expression inside the arctangent. We know that:

\[ 1 - \sin x = \cos^2\left(\frac{x}{2}\right) \]
\[ 1 + \sin x = \cos^2\left(\frac{x}{2}\right) + 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) = \left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)^2 \]

Thus, we can write:

\[ \sqrt{1 - \sin x} = \sqrt{\cos^2\left(\frac{x}{2}\right)} = \cos\left(\frac{x}{2}\right) \]
\[ \sqrt{1 + \sin x} = \sqrt{\left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)^2} = \cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right) \]

Now substituting these into the expression gives:

\[ y = \tan^{-1}\left(\frac{\cos\left(\frac{x}{2}\right) + \left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)}{\cos\left(\frac{x}{2}\right) - \left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)}\right) \]

**Step 2: Further simplification**

This simplifies to:

\[ y = \tan^{-1}\left(\frac{2\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)}{-\sin\left(\frac{x}{2}\right)}\right) \]

This can be rewritten as:

\[ y = \tan^{-1}\left(-\frac{2\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)}\right) \]

**Step 3: Using properties of tangent**

Using the property of tangent, we can express this as:

\[ y = -\tan^{-1}\left(\frac{2\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)} + 1\right) \]

**Step 4: Final simplification**

This leads to:

\[ y = -\left(\frac{\pi}{2} - \frac{x}{2}\right) = \frac{x}{2} - \frac{\pi}{2} \]

Thus, we conclude that:

\[ y = \frac{\pi}{2} - \frac{x}{2} \]

**Final Result:**

The value of \( y \) is:

\[ \frac{\pi}{2} - \frac{x}{2} \]