LHS=1+cot2 α1+cosec α=1+cos2 αsin2 α1+1sin α
[∵cot θ=cos θsin θ and cosec θ=1sin θ]
=1+cos2 αsin α(1+sin α)=sin α(1+sin α)+cos2 αsin α(1+sin α)
=sin α+(sin2 α+cos2 α)sin α(1+sin α)[∵sin2 θ+cos2 θ=1]
=(sin α+1)sinα(sin α+1)=1sin α[∵cosec θ=1sin θ]
=cosec α=RHS