Question

1 over 1+tanx + 1 over 1+cotx=1 verify the identities

Answer 1

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

=sin^2x/ cosx( sinx-cosx)
Similarly
cotx/(1-tanx)= cos^x/sinx(cosx-sinx)
LHS=(tan x / (1 - cot x)) + (cot x / (1 - tan x))
= sin^2x/ cosx( sinx-cosx)+cos^x/sinx(cosx-sinx)
=(sin^3x-cos^x)/(sin x cosx(sinx-cosx))
=(sin^2x+sinx cosx+cos^2x)/ sinx cosx
=( 1+ sinx cosx)/ sinx cosx
= 1/ sinx cosx +1
=1+ 1/sinx cosx
=1+ sec x cosex