Prove that # sqrt((1-cosx)/(1+cosx)) -= (1-cosx)/(|sinx|) # ?

2 Answers
Feb 14, 2018

Please see below.

Explanation:

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#sqrt((1-cosx)/(1+cosx))=sqrt(((1-cosx)(1-cosx))/((1+cosx)(1-cosx)))=#

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

Feb 14, 2018

We seek to prove that:

# sqrt((1-cosx)/(1+cosx)) -= (1-cosx)/(|sinx|) #

Consider the RHS:

# RHS = (1-cosx)/(|sinx|) #

# \ \ \ \ \ \ \ \ = sqrt( ((1-cosx)/(|sinx|))^2 ) #

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

# \ \ \ \ \ \ \ \ = sqrt( (1-cosx)^2/(1-cos^2x) ) #

# \ \ \ \ \ \ \ \ = sqrt( (1-cosx)^2/((1+cosx)(1-cosx) ) #

# \ \ \ \ \ \ \ \ = sqrt( (1-cosx)/(1+cosx) ) #

# \ \ \ \ \ \ \ \ = LHS \ \ \ \ # QED