Question

tan theta / 1-cot theeta + cot theeta / 1- tan theeta = 1+ sec theeta cosec theeta

Answer 1

the answer is as follows.....hope it helps .....
LHS= tanQ/1-cotQ + cotQ/1-tanQ
= (sinQ/cosQ)/1-(cosQ/sinQ)+(cosQ/sinQ)/1-(sinQ/cosQ)
= (sinQ/cosQ)/(sinQ-cosQ)/sinQ)+(cosQ/sinQ)/(cosQ-sinQ)/cosQ
=(sinQ/cosQ)×sinQ/(sinQ-cosQ)+(cosQ/sinQ)×cosQ/(cosQ-sinQ)
=sin^2Q/cosQ(sinQ-cosQ)+ cos^2Q/sinQ(cosQ-sinQ)
=(sin^3Q+(-cos^3Q))/sinQcosQ(sinQ-cosQ)
=(sin^3Q-cos^3Q)/sinQcosQ(sinQ-cosQ)
=(sinQ-cosQ)(sin^2Q+sinQcosQ+cos^2Q)/sinQcosQ(sinQ-cosQ)
=(sin^2Q+sinQcosQ+cos^2Q)/sinQcosQ
=(1+sinQcosQ)/sinQcosQ
=secQcosecQ+1
...