Question

Prove that tan60=2tan30/1-tan^2 30

Answer 1

Step-by-step explanation:

should be help full I think

Answer 2

TO PROVE :-

$\to \sf \: \tan \: 60^{ \circ} = \dfrac{2 \tan \:30^{ \circ}}{1 - \tan^{2} \: 30^{ \circ} }$

GIVEN :-

$\to \sf \: \tan \: 60^{ \circ} = \dfrac{2 \tan \:30^{ \circ}}{1 - \tan^{2} \: 30^{ \circ} }$

SOLUTION :-

➠ Refer to the chart for the value of tan 60° and tan 30°.

$\huge\displaystyle\begin{tabular}{|c|c|c|c|c|c|}\cline{1-6}&0^{\circ}&30^{\circ}&45^{\circ}&60^{\circ}&90^{\circ}\cline{1-6}\cline{1-6}Sin&0& \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} &1\cline{1-6}\cline{1-6}Cos&1& \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} &0\cline{1-6}\cline{1-6}Tan&\infty& \dfrac{1}{\sqrt{3}} &1& \sqrt{3} &0 \cline{1 - 5}\cline{1 - 5}\end{tabular}$

$\to \mathbb{RHS} \\ \\ \to \sf \: \dfrac{2 \tan \:30^{ \circ}}{1 - \tan^{2} \: 30^{ \circ}} \\ \\ \to \sf \: \left( \dfrac{2 \times \dfrac{1}{ \sqrt{3} } }{1 - \bigg(\dfrac{1}{ \sqrt{3} } \bigg)^{2} } \right) \\ \\ \to \sf \: \left( \dfrac{ \dfrac{2}{ \sqrt{3} } }{ 1 - \dfrac{1}{ \sqrt{9} } } \right) \\ \\ \to \sf \: \left( \dfrac{ \dfrac{2}{ \sqrt{3} } }{1 - \dfrac{1}{3} } \right) \\ \\ \to \sf \left( \dfrac{ \dfrac{2}{ \sqrt{3} } }{ \dfrac{3 - 1}{3} } \right) \\ \\ \to \sf \: \frac{ \cancel{2}}{ \cancel{\sqrt{3}}} \times \frac{ \cancel{3}}{ \cancel{2}} \\ \\ \to \sf \: \sqrt{3} \\ \\ \to \sf \: \tan \: 60 ^{ \circ} \\ \\ \to \mathbb{LHS}$

L.H.S = R.H.S

HENCE VERIFIED ✔