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

##cosx/(1-tanx) + sinx/(1-cotx)=sinx+cosx##

##cosx/(1-tanx) + sinx/(1-cotx) = cosx/(1-sinx/cosx) +sinx/(1-cosx/sinx)##
##= cosx/(1-sinx/cosx) +sinx/(1-cosx/sinx)##
##= cosx/((cosx-sinx)/cosx) +sinx/((sinx-cosx)/sinx)##
##= cos^2x/(cosx-sinx) +sin^2x/(sinx-cosx)##
##= cos^2x/(cosx-sinx) -sin^2x/(cosx-sinx)##
##= (cos^2x-sin^2x)/(cosx-sinx)##
##= [(cosx+sinx)cancel(cosx-sinx)]/cancel(cosx-sinx)##
##=sinx+cosx##

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