The equation of the curve through the origin satisfying the equation…

$\frac{d x}{d y}-y \tan x=\sec x$

I.F. = $e^{-\int \tan x d x}=\cos x$

$y \cos x=\int \sec x \cos x d x=x+c$

$y \cos x=x+c$

At (0, $\theta$), y cos x = x               Since c = 0

Hence (2) is the correct answer.