We have to prove, #tan theta/(1-cot theta) + cot theta/(1-tan theta) = 1+tan theta+ cot theta#
Let us take Left Hand Side (L.H.S.)
#rArr tan theta/(1-cot theta) +cot theta/(1-tan theta)#
#rArr [sin theta/cos theta]/[1-cos theta/sin theta] + [cos theta/sin theta]/[1-sin theta/cos theta]#
#rArr [sin theta/cos theta]/[(sin theta-cos theta)/sin theta] + [cos theta/sin theta]/[(cos theta-sin theta)/cos theta]#
#rArr [sin theta/cos theta. sin theta]/(sin theta-cos theta) +[cos theta/sin theta. cos theta]/[-(sin theta-cos theta)]#
#rArr [sin^2 theta/cos theta]/[sin theta-cos theta] - [cos^2 theta/sin theta]/[sin theta-cos theta]#
#rArr [sin^2 theta/cos theta - cos^2 theta/sin theta]/(sin theta-cos theta)#
#rArr [(sin^3 theta - cos^3 theta)/(sin theta cos theta)]/(sin theta-cos theta)#
#rArr [(sin theta-cos theta)(sin^2 theta+sin theta. cos theta+cos^2 theta)]/(sin theta cos theta). 1/(sin theta-cos theta)#
#rArr sin^2 theta/(sin theta cos theta)+(sin theta. cos theta)/(sin theta. cos theta)+cos^2 theta/(sin theta. cos theta)#
#rArr sin theta/cos theta +1+cos theta/sin theta#
#rArr tan theta+1+cot theta# = L. H. S.
