Prove that cosA-sinA+1cosA+sinA-1=cosecA+cotA
Solve for the required proof
Given that cosA-sinA+1cosA+sinA-1=cosecA+cotA
Consider L.H.S
cosA-sinA+1cosA+sinA-1
Dividing the numerator and denominator by sinA, we get,
cosAsinA-sinAsinA+1sinAcosAsinA+sinAsinA-1sinA
=cotA-1+cosecAcotA+1-cosecA ; ∵cotA=cosAsinA,cosecA=1sinA
=cotA-cosec2A-cot2A+cosecAcotA+1-cosecA ;∵1+cot2A=cosec2A
=cotA+cosecA-cosecA+cotAcosecA-cotAcotA+1-cosecA ; ∵a2-b2=a+ba-b
=cotA+cosecA1-cosecA-cotAcotA+1-cosecA
=cotA+cosecAcotA+1-cosecAcotA+1-cosecA
=cotA+cosecA
=R.H.S
⇒L.H.S=R.H.S
Hence, the given identity is proved.