The equation ` "cosec " x/2 + " cosec " y/2 + " cosec " z/2=6`, where `0 lt x, y,z lt pi/2` and `x+y+z=pi`, have
i) Three Ordered triplet (x,y,z)solutions
ii) Two ordered triplet (x,y,z) solutions.
iii) Just one ordered triplet (x,y,z) solution
iv) No ordered triplet (x,y,z) solution

Options

A. three Ordered triplet `(x,y,z)` solutions
B. Two ordered triplet (x,y,z) solutions.
C. Just one ordered triplet (x,y,z) solution
D. No ordered triplet (x,y,z) solution

Answer

To solve the equation \( \csc \frac{x}{2} + \csc \frac{y}{2} + \csc \frac{z}{2} = 6 \) under the conditions \( 0 < x, y, z < \frac{\pi}{2} \) and \( x + y + z = \pi \), we can follow these steps:

### Step 1: Set up the problem
Given the equation and conditions, we know that \( x + y + z = \pi \). This implies that \( z = \pi - x - y \).

### Step 2: Substitute \( z \) in the equation
Substituting \( z \) into the equation gives:
\[ \csc \frac{x}{2} + \csc \frac{y}{2} + \csc \frac{\pi - x - y}{2} = 6 \]
Using the identity \( \csc(\frac{\pi}{2} - \theta) = \sec(\theta) \), we can rewrite \( \csc \frac{\pi - x - y}{2} \) as \( \sec \frac{x + y}{2} \).

### Step 3: Analyze the function behavior
We analyze the function \( \csc \frac{x}{2} \) and \( \csc \frac{y}{2} \) as \( x \) and \( y \) vary from \( 0 \) to \( \frac{\pi}{2} \). The cosecant function approaches infinity as its argument approaches zero and decreases as the argument increases.

### Step 4: Consider equal values for \( x, y, z \)
Assuming \( x = y = z \), we have:
\[ 3x = \pi \implies x = y = z = \frac{\pi}{3} \]
Calculating the left-hand side:
\[ \csc \frac{\pi/3}{2} = \csc \frac{\pi}{6} = 2 \]
Thus,
\[ \csc \frac{x}{2} + \csc \frac{y}{2} + \csc \frac{z}{2} = 3 \times 2 = 6 \]
This is a valid solution.

### Step 5: Check for other possible solutions
Next, we need to check if there are other combinations of \( x, y, z \) that satisfy the equation. We can try different values for \( x \) and \( y \) while ensuring \( z \) remains positive and less than \( \frac{\pi}{2} \).

### Step 6: Conclusion
After testing various combinations, we find that the only solution that satisfies the equation is \( (x, y, z) = \left(\frac{\pi}{3}, \frac{\pi}{3}, \frac{\pi}{3}\right) \). Therefore, there is only one ordered triplet solution.

### Final Answer
The answer is: **Just one ordered triplet (x, y, z) solution**.

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