Using $frac{{{{sin }^4}x}}{2} + frac{{{{cos }^4}x}}{3} = frac{1}{5}$ finding the value of $frac{{{{sin }^8}x}}{8} + frac{{{{cos }^8}x}}{27}$

If $frac{{{{sin }^4}x}}{2} + frac{{{{cos }^4}x}}{3} = frac{1}{5}$, then

(A)${tan ^2}x = frac{2}{3}$

(B)$frac{{{{sin }^8}x}}{8} + frac{{{{cos }^8}x}}{27} = frac{1}{125}$

(C)${tan ^2}x = frac{1}{3}$

(D) $frac{{{{sin }^8}x}}{8} + frac{{{{cos }^8}x}}{27} = frac{2}{125}$

The official answer is A and B.

My approach is as follow

$frac{{{{sin }^4}x}}{2} + frac{{{{cos }^4}x}}{3} = frac{1}{5} Rightarrow frac{{{{sin }^4}x}}{2} + frac{{{{left( {1 – {{sin }^2}x} right)}^2}}}{3} = frac{1}{5}$

$ Rightarrow frac{{{{sin }^4}x}}{2} + frac{{1 + {{sin }^4}x – 2{{sin }^2}x}}{3} = frac{1}{5} Rightarrow 3{sin ^4}x + 2 + 2{sin ^4}x – 4{sin ^2}x = frac{6}{5}$

$ Rightarrow 2 + 5{sin ^4}x – 4{sin ^2}x = frac{6}{5} Rightarrow 10 + 25{sin ^4}x – 20{sin ^2}x = 6 Rightarrow 25{sin ^4}x – 20{sin ^2}x + 4 = 0 Rightarrow {left( {5{{sin }^2}x – 2} right)^2} = 0 Rightarrow {sin ^2}x = frac{2}{5}$

${cos ^2}x = frac{3}{5}$, Hence ${tan ^2}x = frac{2}{3}$,

which is correct as per the official answer

$frac{{{{sin }^2}x}}{2} = frac{1}{5}& frac{{{{cos }^2}x}}{3} = frac{1}{5}$

cubing and adding we get

$frac{{{{sin }^8}x}}{8} + frac{{{{cos }^8}x}}{{27}} = frac{2}{{125}}$ but it is wrong as per the official answer key and the correcct answer is $frac{{{{sin }^8}x}}{8} + frac{{{{cos }^8}x}}{27} = frac{1}{125}$

Where I am making mistake.