Solve the Equation sqrt(3)tg x - sqrt(3)ctg x=2

Solve the Equation
$\sqrt{3}tgx-\sqrt{3}ctgx=2$
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Yareli Hendrix
$\sqrt{3}tgx-\sqrt{3}ctgx=2$
$\sqrt{3}tgx-\frac{\sqrt{3}}{tgx}=2$
tgx=t
$\sqrt{3}t-\frac{\sqrt{3}}{t}=2$
$\sqrt{3}{t}^{2}-2t-\sqrt{3}=0$
$D=4+4\ast \sqrt{3}\ast \left(-\sqrt{3}\right)=4+4\ast 3=16$
${t}_{1}=\frac{2+4}{2\sqrt{3}}=\frac{3}{\sqrt{3}}=\sqrt{3}$
${t}_{2}=\frac{2-4}{2\sqrt{3}}=-\frac{1}{\sqrt{3}}$
$tgx=\sqrt{3}$
$x=arctg\left(\sqrt{3}\right)+\pi \ast n$
$x=\frac{\pi }{3}+\pi \ast n$
$tgx=-\frac{1}{\sqrt{3}}$
$x=arctg\left(-\frac{1}{\sqrt{3}}\right)+\pi \ast k$
$x=-\frac{\pi }{6}+\pi \ast k$
n and k $\in$ Z.