Prove tan2x/1+tan2x=sin2x?

Answer 1

Please be careful with your symbols. I believe you mean to prove

    tan²x
------------ = sin²x
1 + tan²x

written as tan²x / ( 1 + tan²x ) = sin²x

Replace tan x with sin x / cos x to get

( sin²x / cos²x ) / ( 1 + ( sin²x / cos²x ) )

Multiply numerator and denominator by cos²x to get

sin²x / ( cos²x + sin²x ) = sin²x / 1 = sin²x

Answer 2

if by this you mean

(tan(x)^2)/(1 + tan(x)^2) = sin(x)^2

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