`cot^2``pi/6``+cose c``(5pi)/6``+3tan^2``pi/6=6`

Explanation

To prove the equation \( \cot^2\left(\frac{\pi}{6}\right) + \csc\left(\frac{5\pi}{6}\right) + 3\tan^2\left(\frac{\pi}{6}\right) = 6 \), we will convert all trigonometric functions into sine and cosine forms and simplify the expression step by step.

### Step 1: Convert Trigonometric Functions
We start with the left-hand side:
\[ \cot^2\left(\frac{\pi}{6}\right) + \csc\left(\frac{5\pi}{6}\right) + 3\tan^2\left(\frac{\pi}{6}\right) \]

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