Question

$\: show \: that \: \frac{2 \tan30° }{1+ {tan}^{2} 30°} = sin60°$
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Answer 1

To prove :

$\sf \dfrac {2 tan 30^{\circ}}{1 + tan^2 30^{\circ}} \ = \ sin 60^{\circ}$

Proof :

We know that,

$\quad : \implies \sf tan 30^{\circ} \ = \ \dfrac {1}{\sqrt {3}}$

$\quad : \sf \implies \sf \dfrac {2 tan 30^{\circ}}{1 + tan^2 30^{\circ}}$

$\quad : \implies \sf \dfrac{2 \times \bigg( \dfrac {1}{\sqrt {3}} \bigg)}{1 + \bigg( \dfrac {1}{\sqrt {3}}^2 \bigg)}$

$\quad : \implies \sf \dfrac{2 \times \dfrac {1}{\sqrt{3}}}{1 + \dfrac{1}{3}}$

$\quad : \implies \sf \dfrac{\dfrac {2}{\sqrt {3}}}{\dfrac {4}{3}}$

$\quad : \implies \sf \dfrac {2}{\sqrt {3}} \times \dfrac {3}{4}$

$\quad : \implies \sf \dfrac {2}{\sqrt {3}} \times \dfrac {\sqrt{3} \ \sqrt{3}}{4}$

$\quad : \implies \sf \dfrac {\sqrt {3}}{2}$

$\therefore \sf Hence, \ sin 60^{\circ} \ = \ \dfrac {\sqrt {3}}{2}$