How do you prove tanx tan(1/2)x = sec x- 1?

1 Answer
Jul 13, 2018

Please see the proof below

Explanation:

Express tanx and secx in terms of t=tan(x/2)

tanx=(2t)/(1-t^2)

cosx=(1-t^2)/(1+t^2)

secx=1/cosx=(1+t^2)/(1-t^2)

The

LHS=tanx*tan(x/2)

=(2t)/(1-t^2)*t

=(2t^2)/(1-t^2)

RHS=secx-1

=(1+t^2)/(1-t^2)-1

=(1+t^2-1+t^2)/(1-t^2)

=(2t^2)/(1-t^2)

Therefore,

LHS=RHS

QED