Question

Prove that tanx/1-cotx+cotx/1-tanx=1+tanx+cotx

Answer 1

Step-by-step explanation:

LHS

tanx/1-cotx+cotx/1-tanx

=(sinx/cosx)/(sinx-cosx/sinx)+(cosx/sinx)/(cosx-sinx/cosx)

=sin²x/cosx(sinx-cosx)-cos²x/sinx(sinx-cosx)

=sin³x-cos³x/sinxcosx(sinx-cosx)

=(sinx-cosx)(1+sinxcosx)/ sinxcosx(sinx-cosx)

=(1+sinxcosx)/sinxcosx

=secxcosecx+1

RHS

1+tanx+cotx

=1+(sinx/cosx)+(cosx/sinx)

=sinxcosx+sin²x+cos²x/sinxcosx

=sinxcosx+1/sinxcosx

=1+secxcosecx

LHS=RHS