cos2x=1-tan^2x/1+tan^2x

Question

i want to prove that identity

Answer 1

cos(2x)=cos²(x)-sin²(x) is a basic identity.

1-tan²(x)=1-sin²(x)/cos²(x)=(cos²(x)-sin²(x))/cos²(x)=cos(2x)/cos²(x).

sin²(x)+cos²(x)=1, so sin²(x)/cos²(x)+1=1+tan²(x)=1/cos²(x).

(1-tan²(x))/(1+tan²(x))=cos(2x)/cos²​(x)/(1/cos²(x))=cos(2x)(cos²(x)/cos²(x))=cos(2x) QED.