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

• A

  $2(\sec \mathrm{\theta }-\tan \mathrm{\theta })$

• B

  $2\sec \mathrm{\theta }$

• C

  $-2\tan \mathrm{\theta }$

• D

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