Let `f(x)={(sinx+cosx",",0 lt x lt (pi)/(2)),(a",",x=pi//2),(tan^(2)x+"cosec"x",",pi//2 lt x lt pi):}`
Then its odd extension is

Options

A. `{(-tan^(2)x-"cosec"x",",-pi lt x lt -(pi)/(2)),(-a",",x=-(pi)/(2)),(-sinx+cosx",",-(pi)/(2) lt x lt 0 ):}`
B. `{(-tan^(2)x+"cosec"x",",-pi lt x lt -(pi)/(2)),(-a",",x=-(pi)/(2)),(sinx-cosx",",-(pi)/(2) lt x lt 0 ):}`
C. `{(-tan^(2)x+"cosec"x",",-pi lt x lt -(pi)/(2)),(a",",x=-(pi)/(2)),(sinx-cosx",",-(pi)/(2) lt x lt 0 ):}`
D. `{(tan^(2)x+"cosec"x",",-pi lt x lt -(pi)/(2)),(-a",",x=-(pi)/(2)),(sinx+cosx",",-(pi)/(2) lt x lt 0 ):}`

Explanation

To find the odd extension of the given function \( f(x) \), we will follow these steps:

### Step 1: Understand the function definition
The function \( f(x) \) is defined as:
- \( f(x) = \sin x + \cos x \) for \( 0 < x < \frac{\pi}{2} \)
- \( f(x) = a \) for \( x = \frac{\pi}{2} \)
- \( f(x) = \tan^2 x + \csc x \) for \( \frac{\pi}{2} < x < \pi \)

### Step 2: Define the odd extension
The odd extension of a function \( f(x) \) is defined as:
\[ f_{odd}(x) = \begin{cases} f(x) & \text{if } x \geq 0 \\ -f(-x) & \text{if } x < 0 \end{cases} \]

### Step 3: Find \( f(-x) \)
We will evaluate \( f(-x) \) for the different intervals:
1. **For \( -x \) in \( (0, \frac{\pi}{2}) \)** (which means \( x \) is in \( (-\frac{\pi}{2}, 0) \)):
\[ f(-x) = \sin(-x) + \cos(-x) = -\sin x + \cos x \]

2. **For \( -x = -\frac{\pi}{2} \)**:
\[ f(-x) = a \]

3. **For \( -x \) in \( (\frac{\pi}{2}, \pi) \)** (which means \( x \) is in \( (-\pi, -\frac{\pi}{2}) \)):
\[ f(-x) = \tan^2(-x) + \csc(-x) = \tan^2 x - \csc x \]

### Step 4: Substitute into the odd extension formula
Now we substitute \( f(-x) \) into the odd extension formula:
1. **For \( x \) in \( (-\frac{\pi}{2}, 0) \)**:
\[ f_{odd}(x) = -f(-x) = -(-\sin x + \cos x) = \sin x - \cos x \]

2. **For \( x = -\frac{\pi}{2} \)**:
\[ f_{odd}(x) = -f(-x) = -a \]

3. **For \( x \) in \( (-\pi, -\frac{\pi}{2}) \)**:
\[ f_{odd}(x) = -f(-x) = -(\tan^2 x - \csc x) = -\tan^2 x + \csc x \]

### Step 5: Combine the results
The odd extension \( f_{odd}(x) \) can now be summarized as:
\[ f_{odd}(x) = \begin{cases} \sin x - \cos x & \text{if } -\frac{\pi}{2} < x < 0 \\ -a & \text{if } x = -\frac{\pi}{2} \\ -\tan^2 x + \csc x & \text{if } -\pi < x < -\frac{\pi}{2} \end{cases} \]