How do you prove tanx tanx(1/2)=secx-1?

1 Answer
Nghi N
Dec 12, 2017

#tan (x/2) - sin (x/2)/(cos (x/2))#
We know:
#2sin^2 (x/2) = 1 - cos x --> sin (x/2) = +- sqrt((1 - cos x)/2)#
#2cos^2 (x/2) = 1 + cos x --> cos (x/2) = +- sqrt((1 + cos x)/2)#
Therefor,
#tan (x/2) = +- sqrt((1 - cos x)/(1 + cos x))#
Multiply both numerator and denominator by #sqrt(1 + cos x)#
#tan (x/2) = sqrt ((1 - cos x)^2)/sqrt(1 - cos^2 x) = (1 - cos x)/sin x#
Finally,
#tan x.tan (x/2) = (sin x/cos x)((1 - cos x)/(sin x)) = #
#= (1 - cos x)/cos x = sec x - 1#. Proved

Related questions
Impact of this question
805 views around the world
You can reuse this answer
Creative Commons License