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

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Answer 1 · 2018-07-13T11:42:06

Please see the proof below

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,

$LHS=RHS$

$QED$

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