tan^2x - tanx/ tan^2x + tanx = tan2x - sec 2x

Question

Hi plz could u help me with this Q.

How can prove RHS=LHS

Answer 1

(tan^2(x)-tan(x))/(tan^2(x)+tan(x)) =? tan(2x)-sec(2x)

(tan(x)-1)/(tan(x)+1) =? sin(2x)/cos(2x)-1/cos(2x)

(sin(x)/cos(x)-1)/(sin(x)/cos(x)+1) =? (sin(2x)-1)/cos(2x)

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

(sin(x)-cos(x))/(sin(x)+cos(x))
=? (sin^2(x)+cos^2(x)–2sin(x)cos(x))/(sin^2(x)–cos^2(x))

(sin(x)-cos(x))/(sin(x)+cos(x))
=? (sin(x)–cos(x))^2/((sin(x)–cos(x))(sin(x)+cos(x)))

(sin(x)-cos(x))/(sin(x)+cos(x))
= (sin(x)–cos(x))/(sin(x)+cos(x)) √