As simple as it looks

Geometry Level 2

sin ⁡ 2 A 1 + cos ⁡ 2 A \large \dfrac{\sin 2A}{1 + \cos 2A}

Simplify the trigonometric expression above.

cot ⁡ A \cot A tan ⁡ 2 A \tan 2A cot ⁡ 2 A \cot 2A sin ⁡ A \sin A tan ⁡ A \tan A cos ⁡ A \cos A None of these choices

Abhiram Rao by Abhiram Rao , 18, India

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2 solutions

Ikkyu San
Apr 8, 2016

sin ⁡ 2 A 1 + cos ⁡ 2 A = 2 sin ⁡ A cos ⁡ A 1 + 2 cos ⁡ 2 A − 1 = 2 sin ⁡ A cos ⁡ A 2 cos ⁡ A cos ⁡ A = sin ⁡ A cos ⁡ A = tan ⁡ A \dfrac{\sin{2A}}{1+\cos{2A}}=\dfrac{2\sin{A}\cos{A}}{1+2\cos^2{A}-1}=\dfrac{2\sin{A}\cos{A}}{2\cos{A}\cos{A}}=\dfrac{\sin{A}}{\cos{A}}=\boxed{\tan{A}}

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Perfect Ikkyu !

Abhiram Rao - 5 years, 2 months ago
Samarth K
Apr 8, 2016

sin2A/(1+cos2A)= 2sinAcosA/[1+(cos^{2}A-sin^{2}A)
[As sin2A=2sinAcosA and cos2A=cos^{2}A-sin^{2}A]
= 2sinAcosA/(sin^{2}A+cos^{2}A+cos^{2}A-sin^{2}A)
[As sin^{2}A+cos^{2}A=1]
=2sinAcosA/2cos^{2}A
=sinA/cosA= TanA


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