Let $-\frac{\mathrm{\pi }}{6}<\mathrm{\theta }<\frac{-\mathrm{\pi }}{12}$ . Suppose α₁ and ß₁ are the roots of the equation ${\mathrm{x}}^2-2\mathrm{xsec}\mathrm{\theta }+1=0$ and α₂ and ß₂ are the roots of the equation ${\mathrm{x}}^2+2\mathrm{xtan}\mathrm{\theta }-1=0$ . If ${\mathrm{\alpha }}_1>{\mathrm{\beta }}_1\text{ and }{\mathrm{\alpha }}_2>{\mathrm{\beta }}_2\text{ then }{\mathrm{\alpha }}_1+{\mathrm{\beta }}_2$ equals

$\begin{array}{l}{\mathrm{x}}^2-2\mathrm{xsec}\mathrm{\theta }+1=0\\ \Rightarrow \mathrm{x}=\sec \mathrm{\theta }\pm \tan \mathrm{\theta }\text{ and }-\frac{\mathrm{\pi }}{6}<\mathrm{\theta }<-\frac{\mathrm{\pi }}{12}\\ \Rightarrow \sec \left(-\frac{\mathrm{\pi }}{6}\right)>\sec \mathrm{\theta }>\sec \left(-\frac{\mathrm{\pi }}{12}\right)\\ \Rightarrow \sec \left(\frac{\mathrm{\pi }}{6}\right)>\sec \mathrm{\theta }>\sec \left(\frac{\mathrm{\pi }}{12}\right)\\ \text{ and }\tan \left(-\frac{\mathrm{\pi }}{6}\right)<\tan \mathrm{\theta }<\tan \left(-\frac{\mathrm{\pi }}{12}\right)\\ \Rightarrow \tan \left(\frac{\mathrm{\pi }}{6}\right)>-\tan \mathrm{\theta }>\tan \left(\frac{\mathrm{\pi }}{12}\right)\end{array}$
$$\begin{gathered} \because \mathrm{\alpha },\mathrm{\beta }\text{, are the root of }{\mathrm{x}}^2-2\mathrm{xsec}\mathrm{\theta }+1=0\text{ and }{\mathrm{\alpha }}_1>{\mathrm{\beta }}_1 \\ \begin{array}{l}\Rightarrow {\mathrm{\alpha }}_1=\sec \mathrm{\theta }-\tan \mathrm{\theta }\text{ and }{\mathrm{\beta }}_1=\sec \mathrm{\theta }+\tan \mathrm{\theta }\\ \Rightarrow {\mathrm{\alpha }}_2{\mathrm{\beta }}_2\text{ are the roots of }{\mathrm{x}}^2+2\mathrm{xtan}\mathrm{\theta }-1=0\text{ and }{\mathrm{\alpha }}_2>{\mathrm{\beta }}_2\\ \therefore {\mathrm{\alpha }}_2=-\tan \mathrm{\theta }+\sec \mathrm{\theta },{\mathrm{\beta }}_2=-\tan \mathrm{\theta }-\sec \mathrm{\theta }\end{array} \end{gathered}$$
$\text{Here }{\mathrm{\alpha }}_1+{\mathrm{\beta }}_2=-2\tan \mathrm{\theta }.$