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#