How could I prove that this is true?

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2 Answers
VNVDVI
Apr 8, 2018

See below.

Explanation:

#tanx=sinx/cosx,# so we have

#((sinx/cosx)-sinx)/(2tanx)=sin^2(x/2)#

Subtract the terms in the numerator, taking #cosx# as a common denominator.

#((sinx-sinxcosx)/cosx)/(2tanx)=sin^2(x/2)#

Rewrite the denominator in terms of sine and cosine. Cancel out the relevant terms.

#((sinx-sinxcosx)/cancel(cosx))/(2sinx/cancel(cosx))=sin^2(x/2)#

#(sinx-sinxcosx)/(2sinx)=sin^2(x/2)#

Factor the numerator; cancel out the sine.

#(cancelsinx(1-cosx)/(2cancelsinx))=sin^2(x/2)#

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

Recalling that #sin(x/2)=+-sqrt(1/2(1-cosx))#, we see that the left side is indeed the square of this (squaring causes the #+-# to go away, along with the square root).

So,

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

Jade
Apr 8, 2018

Refer to the explanation.

Explanation:

#(tanx-sinx)/(2tanx) = sin ^2 (x/2)#

Start from #color(red)(RHS)#,

#sin^2 (x/2)#

Since, #color(blue)(1-cosx=2sin^2 (x/2)#

#rArr (1-cosx)/2#

Multiply both #color(red)(N^r)# and #color(red)(D^r)# by #color(blue)(tanx)#

#rArr((1-cosx)* tanx)/(2tanx)#

#rArr(tanx-cosx*tanx)/(2tanx)#

Since #color(blue)(tanx= sinx/cosx)#

#rArr (tanx - cancelcosx* sinx/cancelcosx)/(2tanx)#

#rArr (tanx-sinx)/(2tanx)#

This is our #color(red)(LHS).#

#color(white)(aàaaaaaaaaaaaaaaaaaaaasasssasaaaaaaaaaaas ssh hhccdfhjbcfjnbfhjnvdgjjngfjkknbfjkmbghkkoutdssvhkogfvhkoydvbkkgfvbkkhfvbkkufvbnkfbmiggv)#

Hope this helps :)

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