# Verify (2*tan x)/(1 + tan^2 x) = sin 2x

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The relation `(2*tan x)/(1 + tan^2 x) = sin 2x` has to be verified.

`(2*tan x)/(1 + tan^2 x)`

=> `((2*sin x)/cos x)/(1 + (sin^2x)/cos^2x)`

=> `((2*sin x)/cos x)/((cos^2x + sin^2x)/(cos^2x))`

=> `(2*sin x*cos^2 x)/(cos x*(cos^2x + sin^2x))`

=> `2*sin x*cos x`

=> `sin 2x`

**This proves that **`(2*tan x)/(1 + tan^2 x) = sin 2x`

L:H:S ≡ sin 2x

**we know that, sin2θ = 2sinθ.cosθ**

= 2sinx.cosx ÷ 1

= 2sinx.cosx/cos²x ÷ 1/cos²x

= 2tanx ÷ sec²x

**⇒ use the identity, 1 + tan²A = sec²A**

= 2tanx ÷ (1+tan²x)

= R:H:S

R:H:S ≡ sin 2x

**we know that, sin2θ = 2sinθ.cosθ**

= 2sinx.cosx ÷ 1

=(2sinx.cosx/cos²x) ÷ 1/cos²x

= 2tanx ÷ sec²x

**⇒ use the identity, sec²A = 1 + tan²A**

= 2tanx ÷ (1+tan²x)

= R:H:S