Let $a,b$ and $c$ be three non-coplanar vectors and $d$ be $a$ non-zero vector, which is perpendicular to $a+b+c$. Now, if $d=\left(\sin x\right)(a\times b)+\left(\cos y\right)(b\times c)+2(c\times a)$, then minimum value of $x^2+y^2$ is equal to

$a,b,c$ are non-coplanar $\Rightarrow [a,b,c]\ne 1$Also, $a\times b,b\times c,c\times a$ are non-coplanar given
$d=\sin x(a\times b)+\cos y(b\times c)+2(c\times a)$
Taking dot product with $a+b+c$, we get
$O=\sin x\left[abc\right]+\cos y\left[abc\right]+2\left[abc\right]$
$$\begin{gathered} \Rightarrow \sin x+\cos y+2=0 \\ \Rightarrow \sin x+\cos y=-2 \\ \Rightarrow x=(4n-1)\frac{\pi }{2},y=(2n-1)\pi ,n\in z \end{gathered}$$

for least value of $x^2+y^2,x=\frac{-\pi }{2},y=\pi$ and least value is $\frac{5{\pi }^2}{4}$.