How do you prove that # 1- cos^2(x/2) = (sin^2x)/(2(1+cosx)) #?

2 Answers
Jun 5, 2018

Using that
#cos(x)=2cos^2(x/2)-1#

Explanation:

so we have
#2(1+cos(x))=2(2cos^2(x/2)-1)=4cos^2(x/2)#
so we get
#4cos^2(x/2)(1-cos^2(x/2))#
and
#sin^2(x)=1-cos^2(x)#
#1-(2cos^2(x/2)-1)^2#
#1-4cos^2(x/2)-1+4cos^2(x/2)#
#4cos^2(x/2)(1-cos^2(x/2))#

Jun 5, 2018

Please find a Proof in Explanation.

Explanation:

Since, #1-cosx=2sin^2(x/2)#, we have,

#sin^2x/(2(1+cosx))=(1-cos^2x)/(2(1+cosx))#,

#={cancel((1+cosx))(1-cosx)}/(2cancel((1+cosx))#,

#=(1-cosx)/2#,

#={cancel(2)sin^2(x/2)}/cancel(2)#,

#=sin^2(x/2)#,

#=1-cos^2(x/2)#, as desired!