The value of $$\sin^{-1}\cot\left[\sin^{-1}\left\{\frac{1}{2}\left(1-\sqrt{\frac{5}{6}}\right)\right\}+\cos^{-1}\sqrt{\frac{2}{3}}+\sec^{-1}\sqrt{\frac{8}{3}}\right]$$ is:

To solve this problem, let us simplify the expression step by step, starting with the inverse trigonometric terms inside the bracket.

Let us rewrite the secant term into an inverse cosine function using the reciprocal relation:

$$\sec^{-1} \sqrt{\frac{8}{3}} = \cos^{-1} \sqrt{\frac{3}{8}}$$

Now, let us define two angles $$\alpha$$ and $$\beta$$ as:

$$\alpha = \cos^{-1} \sqrt{\frac{2}{3}} \implies \cos \alpha = \sqrt{\frac{2}{3}}$$

$$\beta = \cos^{-1} \sqrt{\frac{3}{8}} \implies \cos \beta = \sqrt{\frac{3}{8}}$$

From these, we can easily find the corresponding sine values:

$$\sin \alpha = \sqrt{1 - \left(\sqrt{\frac{2}{3}}\right)^2} = \sqrt{1 - \frac{2}{3}} = \frac{1}{\sqrt{3}}$$

$$\sin \beta = \sqrt{1 - \left(\sqrt{\frac{3}{8}}\right)^2} = \sqrt{1 - \frac{3}{8}} = \sqrt{\frac{5}{8}}$$

Next, let us evaluate the cosine of the sum of these two angles, $$\cos(\alpha + \beta)$$:

$$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$$

$$\cos(\alpha + \beta) = \left(\sqrt{\frac{2}{3}}\right)\left(\sqrt{\frac{3}{8}}\right) - \left(\frac{1}{\sqrt{3}}\right)\left(\sqrt{\frac{5}{8}}\right)$$

$$\cos(\alpha + \beta) = \sqrt{\frac{6}{24}} - \sqrt{\frac{5}{24}} = \frac{\sqrt{6} - \sqrt{5}}{2\sqrt{6}}$$

Rationalizing or simplifying the expression:

$$\cos(\alpha + \beta) = \frac{1}{2} \left(1 - \sqrt{\frac{5}{6}}\right)$$

Thus, we have:

$$\alpha + \beta = \cos^{-1}\left[ \frac{1}{2} \left(1 - \sqrt{\frac{5}{6}}\right) \right]$$

We know the standard co-function identity relating arcsine and arccosine:

$$\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$$

Let $$x = \frac{1}{2} \left(1 - \sqrt{\frac{5}{6}}\right)$$.

Then the first term in our main expression is $$\sin^{-1}(x)$$, and the sum of the remaining two terms is $$\alpha + \beta = \cos^{-1}(x)$$.

Therefore, the entire argument inside the cotangent function simplifies to:

$$\sin^{-1}\left\{ \frac{1}{2} \left(1 - \sqrt{\frac{5}{6}}\right) \right\} + \cos^{-1} \sqrt{\frac{2}{3}} + \sec^{-1} \sqrt{\frac{8}{3}} = \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$$

Substituting this back into the original expression:

$$\sin^{-1} \cot\left(\frac{\pi}{2}\right)$$

Since $$\cot\left(\frac{\pi}{2}\right) = 0$$, the expression reduces to:

$$\sin^{-1}(0) = 0$$

The correct option is A.