Question

How do you prove `(sin 2x) / (1 + cos2x) = tan x`?

Answer 1

see explanation

Explanation:

Manipulating the left side using`color(blue)" Double angle formulae "`

`• sin2x = 2sinxcosx`

`• cos2x = cos^2x - sin^2x`

and using `sin^2x + cos^2x = 1 " we can also obtain "`

`cos2x = (1 - sin^2x) - sin^2x = 1 - 2sin^2x`

and `cos2x = cos^2x - (1 - cos^2x ) = 2cos^2x - 1`

`rArr(sin2x)/(1+cos2x) = (2sinxcosx)/(1+2cos^2x-1) = (2sinxcosx)/(2cos^2x)`

`= (cancel(2) sinx cancel(cosx))/(cancel(2) cancel(cosx) cosx)= (sinx)/(cosx) = tanx = " right side "`