Separate equations of the pair of lines $x^2+2xy\sec\theta+y^2=0$

Question

Find the separate equations of the lines $x^2+2xy\sec\theta+y^2=0$

Attempt 1

$$ x^2+2xy\sec\theta+y^2=0\\ \frac{x^2}{y^2}+2\frac{x}{y}\sec\theta+1=0\\ \frac{x}{y}=\frac{-2\sec\theta\pm\sqrt{4\sec^2\theta-4}}{2}=-\sec\theta\pm\tan\theta=\frac{-1\pm\sin\theta}{\cos\theta}\\ x\cos\theta=-y(1\pm\sin\theta)\\ x\cos\theta+y(1\pm\sin\theta)=0 $$

Attempt 2

$$ x^2+2xy\sec\theta+y^2=0\\ \frac{y^2}{x^2}+2\frac{y}{x}\sec\theta+1=0\\ \frac{y}{x}=\frac{-2\sec\theta\pm\sqrt{4\sec^2\theta-4}}{2}=-\sec\theta\pm\tan\theta=\frac{-1\pm\sin\theta}{\cos\theta}\\ y\cos\theta=-x(1\pm\sin\theta)\\ y\cos\theta+x(1\pm\sin\theta)=0 $$

Why do I seems to get different solutions for the lines in attempts 1 and 2 ?

Answer 1

\begin{align} \dfrac{-1 \pm \sin \theta}{\cos \theta} &= \dfrac{-1 \pm \sin \theta}{\cos \theta} \cdot \dfrac{-1 \mp \sin \theta}{-1 \mp \sin \theta} \\ &= \dfrac{1-\sin^2 \theta}{\cos \theta (-1 \mp \sin \theta)} \\ &= \dfrac{\cos^2 \theta}{\cos \theta (-1 \mp \sin \theta)} \\ &= \dfrac{\cos \theta}{-1 \mp \sin \theta} \\ \end{align}