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Solve the equation $\frac{1}{\left(\mathrm{cos}\theta {\right)}^{2}}=2\sqrt{3}\mathrm{tan}\theta -2$
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Your equation can be written as
$\frac{1}{{\mathrm{cos}}^{2}\left(x\right)}=$
$1+{\mathrm{tan}}^{2}\left(x\right)=2\sqrt{3}\mathrm{tan}\left(x\right)-2$
or
${\mathrm{tan}}^{2}\left(x\right)-2\sqrt{3}\mathrm{tan}\left(x\right)+3=0$
the reduced discriminant is
$\delta =3-3=0$
thus, there is one solution given by
$\mathrm{tan}\left(x\right)=\sqrt{3}$ which gives
$x=\frac{\pi }{3}.$