The value of `(sqrt(1 + sin x )+sqrt(1 - sin x))/(sqrt(1 + sin x )-sqrt(1-sin x))`is equal to

Options

A. cosec x + cot x
B. cosec x + tan x
C. sec x + tan x
D. cosec x - cot x

Answer

To solve the expression \(\frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}\), we can use the method of component and dividend. Let's go through the steps one by one.

### Step 1: Apply Component and Dividend
We can express the given fraction using the component and dividend method. According to this method, we can rewrite the expression as:

\[ \frac{(a + b)}{(a - b)} = \frac{1 + \frac{b}{a}}{1 - \frac{b}{a}} \]

where \(a = \sqrt{1 + \sin x}\) and \(b = \sqrt{1 - \sin x}\).

### Step 2: Rewrite the Expression
Using the above method, we can write:

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

### Step 3: Simplify the Denominator
The denominator simplifies as follows:

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

### Step 4: Simplify the Numerator
Now, let's simplify the numerator:

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

This simplifies to:

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

### Step 5: Combine the Results
Now we can combine the results from the numerator and denominator:

\[ \frac{2 + 2\cos x}{2\sin x} = \frac{2(1 + \cos x)}{2\sin x} = \frac{1 + \cos x}{\sin x} \]

### Step 6: Final Simplification
We can further simplify this to:

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

### Final Answer
Thus, the value of the expression is:

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

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