Question

if A=30°,then prove that sin 2A=2sinAcosA=(2tanA)/(1+tan^2A​

Answer 1

GIVEN :

A = 30°

To prove :

sin2A = 2sinAcosA = (2tanA) ÷ (1+tan²A)

Solution :

We will have to find individual value of sin2A , 2sinAcosA & (2tanA) ÷ (1+tan²A) .

➝ sin 2A

= sin 2(30°)

= sin 60°

= (√3)/2

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➝ 2sinA cosA

= 2 × (sin 30°) × (cos 30°)

= 2 × (1/2) × (√3)/2

= (√3)/2

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➝ (2tanA) ÷ (1+tan²A)

$\sf \implies \: \dfrac{2 \times \tan( {30}^{ \circ} ) }{1 + { \tan( {30}^{ \circ} ) }^{2} } \\ \\ \sf \implies \: \dfrac{2 \times \dfrac{1}{ \sqrt{3} } }{1 + { \bigg \{ \: \dfrac{1}{ \sqrt{3} } \bigg\} }^{2} } \\ \\ \sf \implies \: \dfrac{ \dfrac{2}{ \sqrt{3} } }{1 + \dfrac{1}{3} } \\ \\ \sf \implies \: \dfrac{ \dfrac{2}{ \sqrt{3} } }{ \dfrac{(3 + 1)}{3} } \\ \\ \sf \implies \: \dfrac{2}{ \sqrt{3} } \times \dfrac{3}{4} \\ \\ \sf \implies \: \dfrac{ \sqrt{3} }{2}$

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From above , it is clear that ,

sin2A = 2 sinA cosA = (2tanA) ÷ (1+tan²A) = (√3)/2

Hence PROVED.