Question

2tan 30º/(1+tan^2 30º)


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Answer 1

1 2

I hope this helped you a lot

Answer 2

GIVEN :-

• (2 tan 30°) / (1 + tan² 30°)

TO FIND :-

• The value of (2 tan 30°) / (1 + tan² 30°)

SOLUTION :-

☛ Consider the trigonometry ratio's Chart.

$\Large{ \begin{tabular}{|c|c|c|c|c|c|} \cline{1-6} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 65^{\circ} & \sf 90^{\circ} \\ \cline{1-6} \sin & 0 & \dfrac{1}{2 } & \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} & 1 \\ \cline{1-6} \cos & 1 & \dfrac{ \sqrt{ 3 }}{2} } & \dfrac{1}{ \sqrt{2} } & \dfrac{ 1 }{ 2 } & 0 \\ \cline{1-6} \tan & 0 & \dfrac{1}{ \sqrt{3} } & 1 & \sqrt{3} & \infty \\ \cline{1-6} \cot & \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } &0 \\ \cline{1 - 6} \sec & 1 & \dfrac{2}{ \sqrt{3}} & \sqrt{2} & 2 & \infty \\ \cline{1-6} \csc & \infty & 2 & \sqrt{2 } & \dfrac{ 2 }{ \sqrt{ 3 } } & 1 \\ \cline{1 - 6}\end{tabular}}$

$\mapsto \sf \: \dfrac{(2 \: \tan 30 ^{ \circ} ) }{(1 + \tan ^{2} 30^{\circ })}$

☛ Now Substitute the values from above table.

$\mapsto \sf \dfrac{ \bigg \lgroup2 \times \dfrac{1}{ \sqrt{3} } \bigg \rgroup}{1 + \bigg \lgroup \dfrac{1}{ \sqrt{3} } \bigg \rgroup^{2} } \\ \\ \\ \mapsto \sf \: \dfrac{ \bigg(\dfrac{2}{ \sqrt{3}} \bigg) }{ \bigg(1 + \dfrac{1}{ \sqrt{9} } \bigg) } \\ \\ \\ \mapsto \sf \: \dfrac{ \bigg(\dfrac{2}{ \sqrt{3} } \bigg) }{ \bigg(1 + \dfrac{1}{3} \bigg)} \\ \\ \\ \mapsto \sf \: \dfrac{ \bigg(\dfrac{2}{ \sqrt{3} } \bigg)}{ \bigg(\dfrac{4}{3} \bigg)} \\ \\ \\ \mapsto \sf \: \dfrac{ \cancel2}{ \sqrt{ \cancel3} } \times\dfrac{ \cancel3}{ \cancel4} \\ \\ \\ \mapsto \underline{ \boxed{ \red{ \bf \: \dfrac{(2 \: \tan 30 ^{ \circ} ) }{(1 + \tan ^{2} 30^{\circ })} = \dfrac{ \sqrt{3} }{2}}}}$

Hence the required answer is √3/2.