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The question involves trigonometric identities and their implications on triangle geometry. We're given:

1. \(A, B, C\) are non-zero acute angles.
2. The equation \(\sum (\tan A - \cot A) = \sum \tan A - 2 \sum \cot A\).
3. If the minimum value of \(\sum \left(\frac{{\tan^2 A + \cot A}}{2}\right)\) exists, \(A, B, C\) will form a specific type of triangle (scalene, right, equilateral, or isosceles).

**Step-by-step Solution:**

### Analyzing the First Condition
Given the identity:
\[
\sum (\tan A - \cot A) = \sum \tan A - 2 \sum \cot A
\]
This holds true due to the distributive property of summation over subtraction.

### Understanding the Second Condition
We analyze:
\[
\sum \left(\frac{{\tan^2 A + \cot A}}{2}\right)
\]
We need to find the minimum value of this expression to determine the type of triangle. Using the trigonometric identity for \(\cot A\), we can rewrite it in terms of \(\tan A\):
\[
\cot A = \frac{1}{\tan A}
\]
Thus:
\[
\tan^2 A + \

If A, B, C are non-zero acute angles and Σ(tanA / tanB + tanC) = Σ(tanA) - 2Σ(cotA) and if the minimum value of Σ(tan^2(A/2) + cotA) exists, A, B, C will form a triangle.

Solve the trigonometric problem involving acute angles A, B, and C. Learn how the sum of tangent and cotangent expressions leads to identifying the type of triangle formed. This problem is suitable for students in Grades 10-12.

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