y = sin (sin x)
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = cos (sin x) . cos x
$\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}$ = cos (sin x) - . (- sin x) + cos x . (- si (sin x). cos x)
$\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}$ = - sin x . cos (sin x) - y $\mathit{c}\mathit{o}{\mathit{s}}^2$ x
now,
$\frac{{\mathit{d}}^2\mathit{y}}{\mathit{d}{\mathit{x}}^2}$ + tan x $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + y $\mathit{c}\mathit{o}{\mathit{s}}^2$ x
= - sin x . cos (sin x) - y $\mathit{c}\mathit{o}{\mathit{s}}^2$ x + tan x . cos (sin x) . cos x + y $\mathit{c}\mathit{o}{\mathit{s}}^2$ x
= - sin x . cos (sin x) - y $\mathit{c}\mathit{o}{\mathit{s}}^2$ x + $\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$ . cos (sin x) . cos x + y $\mathit{c}\mathit{o}{\mathit{s}}^2$ x
= 0