Step-by-step Solution:

1. Simplify the given trigonometric equation. The given equation is: ${{{{\sin }^4}x} \over 2} + {{{{\cos }^4}x} \over 3} = {1 \over 5}$ To solve this, we can express the equation in terms of a single trigonometric function. Let $s = \sin^2 x$. Then, using the identity $\sin^2 x + \cos^2 x = 1$, we have $\cos^2 x = 1 - s$. The equation can be rewritten as: $\frac{(\sin^2 x)^2}{2} + \frac{(\cos^2 x)^2}{3} = \frac{1}{5}$ Substituting $s$ into the equation: $\frac{s^2}{2} + \frac{(1-s)^2}{3} = \frac{1}{5}$

2. Solve the resulting algebraic equation for $s$. To clear the denominators, we multiply the entire equation by the least common multiple of 2, 3, and 5, which is 30. $30 \left( \frac{s^2}{2} \right) + 30 \left( \frac{(1-s)^2}{3} \right) = 30 \left( \frac{1}{5} \right)$ $15s^2 + 10(1-s)^2 = 6$ Expand the term $(1-s)^2$: $15s^2 + 10(1 - 2s + s^2) = 6$ $15s^2 + 10 - 20s + 10s^2 = 6$ Combine like terms to form a quadratic equation: $25s^2 - 20s + 4 = 0$ This is a perfect square trinomial, which can be factored as: $(5s - 2)^2 = 0$ This gives a single solution for $s$: $5s - 2 = 0 \implies s = \frac{2}{5}$ Since $s = \sin^2 x$, we have $\sin^2 x = \frac{2}{5}$.

3. Find the values of $\cos^2 x$ and $\tan^2 x$. Using the value of $\sin^2 x$, we find $\cos^2 x$: $\cos^2 x = 1 - \sin^2 x = 1 - \frac{2}{5} = \frac{3}{5}$ Now, we can find $\tan^2 x$: $\tan^2 x = \frac{\sin^2 x}{\cos^2 x} = \frac{2/5}{3/5} = \frac{2}{3}$

4. Evaluate each of the given options.

   • A: ${\tan ^2}x = {2 \over 3}$ Our calculated value for $\tan^2 x$ is exactly $\frac{2}{3}$. Therefore, option A is correct.

   • B: ${{{{\sin }^8}x} \over 8} + {{{{\cos }^8}x} \over {27}} = {1 \over {125}}$ Let's calculate the left-hand side (LHS) using our values for $\sin^2 x$ and $\cos^2 x$. $\sin^8 x = (\sin^2 x)^4 = \left(\frac{2}{5}\right)^4 = \frac{16}{625}$ $\cos^8 x = (\cos^2 x)^4 = \left(\frac{3}{5}\right)^4 = \frac{81}{625}$ Now substitute these into the expression: $\text{LHS} = \frac{16/625}{8} + \frac{81/625}{27} = \frac{16}{625 \cdot 8} + \frac{81}{625 \cdot 27} = \frac{2}{625} + \frac{3}{625} = \frac{5}{625} = \frac{1}{125}$ The right-hand side (RHS) is $\frac{1}{125}$. Since LHS = RHS, option B is correct.

   • C: ${\tan ^2}x = {1 \over 3}$ Our calculated value is $\tan^2 x = \frac{2}{3}$. This contradicts option C. Therefore, option C is incorrect.

   • D: ${{{{\sin }^8}x} \over 8} + {{{{\cos }^8}x} \over {27}} = {2 \over {125}}$ From our calculation for option B, the value of the expression is $\frac{1}{125}$. This contradicts option D. Therefore, option D is incorrect.

Conclusion:

Based on the calculations, options A and B are correct.