If `x,y,z` be all positive acute angles then the least value of `tanx(coty+cotz)+tany(cotz+cotx)+tanz(cotx+coty)` is

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To find the least value of the expression \( E = \tan x (\cot y + \cot z) + \tan y (\cot z + \cot x) + \tan z (\cot x + \cot y) \), where \( x, y, z \) are all positive acute angles, we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-step Solution:

1. **Rewrite the Expression**:
\[ E = \tan x \cot y + \tan x \cot z + \tan y \cot z + \tan y \cot x + \tan z \cot x + \tan z \cot y \]

2. **Group Terms**:
We can group the terms in pairs:
\[ E = (\tan x \cot y + \tan y \cot x) + (\tan x \cot z + \tan z \cot x) + (\tan y \cot z + \tan z \cot y) \]

3. **Apply AM-GM Inequality**:
For each pair, we apply the AM-GM inequality:
\[ \tan x \cot y + \tan y \cot x \geq 2 \sqrt{\tan x \cot y \tan y \cot x} \]
This can be simplified to:
\[ = 2 \sqrt{1} = 2 \]
because \( \tan x \cot x = 1 \).

4. **Sum the Inequalities**:
Applying the AM-GM inequality to each of the three pairs gives:
\[ E \geq 2 + 2 + 2 = 6 \]

5. **Conclusion**:
Thus, the least value of \( E \) is \( 6 \).

### Final Answer:
The least value of \( \tan x (\cot y + \cot z) + \tan y (\cot z + \cot x) + \tan z (\cot x + \cot y) \) is \( 6 \).

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