If tg^2alpha=3 then sin^2(2alpha) equal ...?

If $t {g}^{2} \alpha = 3$ then ${\sin}^{2} \left(2 \alpha\right)$ equal ...? A) $\frac{3}{4}$ B) $\frac{1}{2}$ C) $\frac{1}{4}$ D) $\frac{1}{8}$

Apr 10, 2017

Given

${\tan}^{2} \alpha = 3$

$\implies \tan \alpha = \pm \sqrt{3}$

Now

$\sin 2 \alpha = \frac{2 \tan \alpha}{1 + {\tan}^{2} \alpha}$

$\implies \sin 2 \alpha = \frac{\pm 2 \sqrt{3}}{1 + {\left(\pm \sqrt{3}\right)}^{2}}$

$\implies \sin 2 \alpha = \frac{\pm 2 \sqrt{3}}{4} = \pm \frac{\sqrt{3}}{2}$

So

${\sin}^{2} 2 \alpha = {\left(\pm \frac{\sqrt{3}}{2}\right)}^{2} = \frac{3}{4}$