The angle between the lines represented by cos θ ( cos θ + 1 ) x 2 - 2 cos θ + sin 2 θ x…

We have the equation of pair of straight lines $\cos \theta (\cos \theta +1)x^2-\left(2\cos \theta +{\sin }^2\theta \right)xy+\left(1-\cos \theta \right)y^2=0$

So, angle between them is

$\theta ={\tan }^{-1}\left(\frac{2\sqrt{{\left(\cos \theta +{\displaystyle \frac{{\sin }^2\theta }{2}}\right)}^2-\cos \theta \left(\cos \theta +1\right)\left(1-\cos \theta \right)}}{\cos \theta \left(\cos \theta +1\right)+1-\cos \theta }\right)$

$={\tan }^{-1}\left(\frac{2\sqrt{{\cos }^2\theta +{\displaystyle \frac{{\sin }^4\theta }{4}}+\cos \theta ·{\sin }^2\theta -\cos \theta \left(1-{\cos }^2\theta \right)}}{{\cos }^2\theta +1}\right)={\tan }^{-1}\left(\frac{2\sqrt{{\cos }^2\theta +{\displaystyle \frac{{\sin }^4\theta }{4}}+\cos \theta ·{\sin }^2\theta -\cos \theta {\sin }^2\theta }}{{\cos }^2\theta +1}\right)$

$={\tan }^{-1}\left(\frac{2\sqrt{{\cos }^2\theta +{\displaystyle \frac{{\sin }^4\theta }{4}}}}{{\cos }^2\theta +1}\right)={\tan }^{-1}\left(\frac{\sqrt{4{\cos }^2\theta +{\displaystyle {\sin }^4\theta }}}{{\cos }^2\theta +1}\right)={\tan }^{-1}\left(\frac{\sqrt{4{\cos }^2\theta +{\displaystyle {\left(1-{\cos }^2\theta \right)}^2}}}{{\cos }^2\theta +1}\right)$

$={\tan }^{-1}\left(\frac{\sqrt{4{\cos }^2\theta +{\displaystyle 1+{\cos }^4\theta -2{\cos }^2\theta }}}{{\cos }^2\theta +1}\right)=={\tan }^{-1}\left(\frac{\sqrt{{\left(1+{\cos }^2\theta \right)}^2}}{{\cos }^2\theta +1}\right)={\tan }^{-1}\left(1\right)$

$=\frac{\mathrm{\pi }}{4}$