Prove that tanAsecA+1+cotAcosecA+1=cosecA+secA-cosecA×secA.
STEP 1 : Solving the Left Hand Side (LHS) of the equation
Taking the LHS and solving we get,
tanAsecA+1+cotAcosecA+1
=tanAsecA+1×secA-1secA-1+cotAcosecA+1×cosecA-1cosecA-1
=tanAsecA-1sec2A-1+cotAcosecA-1cosec2A-1
=tanAsecA-1tan2A+cotAcosecA-1cot2A
=secA-1tanA+cosecA-1cotA
=secAtanA-1tanA+cosecAcotA-1cotA
=1cosAsinAcosA-1tanA+1sinAcosAsinA-1cotA
=1sinA-cosAsinA+1cosA-sinAcosA
=cosecA-cosAsinA+secA-sinAcosA
=cosecA+secA-cosAsinA-sinAcosA
=cosecA+secA-cosAsinA+sinAcosA
=cosecA+secA-cos2A+sin2AsinA×cosA
=cosecA+secA-1sinA×cosA
=cosecA+secA-cosecA×secA
i.e. tanAsecA+1+cotAcosecA+1=cosecA+secA-cosecA×secA
Hence proved.