Question

Prove that 1-tan^2x/1+tan^2x = cos2x is an identity?

Answer 1

Please see below.

Explanation:

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

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

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

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

= `cos^2x-sin^2x`

= `cos2x`

Answer 2

`RHS=cos2x`

`=cos(x+x)`

`=cosx*cosx-sinx*sinx`

`=cos^2x-sin^2x`

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

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

`=(1-tan^2x)/(1+tan^2x)=LHS`