How do you prove#tanx/(1+cosx) + sinx/(1-cosx)=cotx+(secx)(cscx)#?

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Answer 1 · 2016-07-16T07:30:58

To prove #tanx/(1+cosx) + sinx/(1-cosx)=cotx+(secx)(cscx)#

#LHS=(sinx(1-cosx))/(cosx(1+cosx)(1-cosx)) + (sinx(1+cosx))/((1-cosx)(1+cosx))#

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

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

#=(1-cosx)/(cosxsinx)+ (cosx(1+cosx))/(sinxcosx)#

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

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

#=(1+cos^2x)/(sinxcosx)#

#=cos^2x/(sinxcosx)+1/(sinxcosx)#

#=cosx/sinx+1/(sinxcosx)#

#=cotx+cscxsecx=RHS#

Proved