Question

Please help me !! Demonstrate the identity

Answer 1

Prove:
`(1 - cos(2x) + sin(2x))/(1 + cos(2x) + sin(2x)) = tan(x)`


`LHS = (1 - cos(2x) + sin(2x))/(1 + cos(2x) + sin(2x))`


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

`sin(2x) = 2sinxcosx`


`LHS = (1 - 1 + 2sin^(2)(x) + 2sinxcosx)/(1 + 2cos^(2)(x) - 1 + 2sinxcosx)`

`= (2sinx(sinx + cosx))/(2cosx(cosx + sinx))`

`= (2sinx)/(2cosx)`

`= tanx = RHS`


`\therefore (1 - cos(2x) + sin(2x))/(1 + cos(2x) + sin(2x)) = tan(x)`