Complex numbers question. Help please?

Question

Prove that #(1-cosx+2icosx)^(-1)=(1-2icot(x/2))/(5+3cosx)#?

So you don't have to retype the equation, without the hashtags it looks like this:

(1-cosx+2icosx)^(-1)=(1-2icot(x/2))/(5+3cosx)

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Answer 1 · 2017-10-15T13:13:28

Please refer to the Explanation. Kindly, check the Problem.

If we substitute #x=pi/2,# then,

#"The L.H.S.="(1-cosx+2icosx)^-1,#

#={1-cos(pi/2)+2icos(pi/2)}^-1,#

#={1-0+2i(0)}^-1,#

#=1^-1,#

#=1,# while, for, #x=pi/2,#

#"The R.H.S.="(1-2icot(x/2))/(5+3cosx),#

#={1-2icot((pi/2)/2)}/(5+3cos(pi/2)),#

#=(1-2icot(pi/4))/(5+3(0)),#

#=(1-2i*1)/5,#

#=(1-2i)/5.#

#"Therefore, The L.H.S."!="The R.H.S."#

Kindly, check the Problem.