Question

( tanx/1-cotx )+ (cotx/1-tanx)= (1+secxcscx)

Good one!

Generally these are done by changing everything to sines and cosines, unless you see some obvious identities.
Also generally, it is best to start with the more complicated side and try to change it to the "easier" side.

so here goes....

LS=
sinx/cosx ÷ (1-cosx/sinx) + cosx/sinx ÷(1-sinx/cosx)
=sinx/cosx [sinx/(sinx-cosx) + cosx/sinx [cosx/(cosx-sinx)
=sin^2 x/(cosx(sinx-cosx)) - cos^2 x/(sinx(sinx-cosx))
= (sin^3x - cos^3 x)/(sinxcosx(sinx-cosx))
=(sinx-cosx)(sin^2x + sinxcosx + cos^2)÷[sinxcosx(sinx-cosx)] difference of cubes!!!
= (sin^2x + cos^2x + sinxcosx)/(sinxcosx)
= (1 + sinxcosx)/sinxcosx

R.S.= 1+ 1/(sinxcosx)
=(sinxcosx + 1)/(sinxcosx) = L.S

Wow!!
=

thank you very much for your help .