$\sin^2x\ge 0\Leftrightarrow 2\sin^2x\ge 0\Leftrightarrow 2\sin^2x+1\ge 1$
$\Leftrightarrow y=\dfrac{4}{1+2\sin^2x}\le 4$
$\to \max y=4$
$\sin^2x\le 1\Leftrightarrow 2\sin^2x\le 2\Leftrightarrow 2\sin^2x+1\le 3$
$\Leftrightarrow y=\dfrac{4}{1+2\sin^2x}\ge \dfrac{4}{3}$
$\to \min y=\dfrac{4}{3}$