Question

Prove that :
sin(2×30°) = 2tan30°/1+tan^2 30°​

Answer 1

$\mathfrak{\underline{\underline{Question:}}}$

Prove that $\: \: \sin(2 \times 30 \degree) = \dfrac{2 \tan30 \degree }{1 + { \tan }^{2}30 \degree }$

$\mathfrak{\underline{\underline{Answer:}}}$

LHS = sin(2 × 30°)

$\qquad$ = sin 60°

$\qquad = \dfrac{ \sqrt{3} }{2}$

$\rm{RHS =\dfrac{2tan 30°}{1+tan^2 30°} } \\ \\ \qquad \: = \dfrac{2( \frac{1}{ \sqrt{3} } )}{1 + {( \frac{1}{ \sqrt{3} }) }^{2} } \\ \qquad \: = \dfrac{ \frac{2}{ \sqrt{3} } }{1 + \frac{1}{3} } \\ \qquad \: = \dfrac{ {2} }{ \sqrt{ 3} } \times \frac{3}{4} \\ \qquad \: = \dfrac{3}{2 \sqrt{3} } \\ \qquad \: = \frac{ \sqrt{3} \times \cancel{\sqrt{3}}}{2 \times \cancel{ \sqrt{3} }} = \frac{ \sqrt{3} }{2}$

∴ LHS = RHS

∴ $\sin(2 \times 30 \degree) = \dfrac{2 \tan30 \degree }{1 + { \tan }^{2}30 \degree }$

Answer 2

Here is your answer. Hope it helps