If `x+y+z =pi //2`, then` cot x +cot y +cot z=`

Options

A. tan x tan y tan z
B. cot x cot y cot z
C. 1
D. `oo`

Answer

To solve the problem, we need to find the value of \( \cot x + \cot y + \cot z \) given that \( x + y + z = \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x + y + z = \frac{\pi}{2} \]

2. **Use the cotangent addition formula:**
We know that:
\[ \tan(x + y + z) = \tan\left(\frac{\pi}{2}\right) \]
Since \( \tan\left(\frac{\pi}{2}\right) \) is undefined (or can be considered as approaching infinity), we can express this using the tangent addition formula:
\[ \tan(x + y + z) = \frac{\tan x + \tan y + \tan z}{1 - \tan x \tan y \tan z} \]
Setting this equal to infinity gives us:
\[ 1 - \tan x \tan y \tan z = 0 \]
Thus,
\[ \tan x \tan y \tan z = 1 \]

3. **Convert tangent to cotangent:**
Recall that \( \tan \theta = \frac{1}{\cot \theta} \). Therefore, we can rewrite the equation:
\[ \frac{1}{\cot x} \cdot \frac{1}{\cot y} \cdot \frac{1}{\cot z} = 1 \]
This implies:
\[ \cot x \cdot \cot y \cdot \cot z = 1 \]

4. **Relate cotangents to their sum:**
We can use the identity:
\[ \cot x + \cot y + \cot z = \frac{\cot x \cot y + \cot y \cot z + \cot z \cot x}{\cot x \cot y \cot z} \]
Since we have \( \cot x \cot y \cot z = 1 \), we can simplify this to:
\[ \cot x + \cot y + \cot z = \cot x \cot y + \cot y \cot z + \cot z \cot x \]

5. **Using the identity for \( \cot x + \cot y + \cot z \):**
From the previous steps, we know:
\[ \cot x + \cot y + \cot z = \cot x \cot y + \cot y \cot z + \cot z \cot x \]
We can conclude that:
\[ \cot x + \cot y + \cot z = 1 \]

### Final Result:
Thus, we have:
\[ \cot x + \cot y + \cot z = 1 \]