if

$y^{2}+2 \log _{e}(\cos x)=y$........(1)

$\Rightarrow 2 y y^{\prime}-2 \tan x=y^{\prime}$.....(2)

From(1), $y(0)=0$ or 1

$\therefore y^{\prime}(0)=0$

Again differentiating (2) we get,

$2\left(y^{\prime}\right)^{2}+2 y y^{\prime \prime}-2 \sec ^{2} x=y^{\prime \prime}$

Put $x=0, y(0)=0,1$ and $y^{\prime}(0)=0$,

we get, $\left|y^{\prime \prime}(0)\right|=2$