Question

How do you prove `(1-tanx)/(1+tanx) = (1-sin2x)/(cos2x)`?

Answer 1

Use the following facts:
`tanx=sinx/cosx`
`sin2x=2sinxcosx`
`cos2x=cos^2x-sin^2x`
`sin^2x+cos^2x=1`

So you get:

`(1-sinx/cosx)/(1+sinx/cosx)=(1-2sinxcosx)/(cos^2x-sin^2x)`
`(cosx-sinx)/cancel(cosx)*cancel((cosx))/cancel((cosx+sinx))=(1-2sinxcosx)/((cosx-sinx)cancel((cosx+sinx)))`

Taking: `(cosx-sinx)` to the left side and using the fact that: `sin^2x+cos^2x=1`

`(cosx-sinx)^2=sin^2x+cos^2x-2sinxcosx` which are indeed equals.