$$\eqalign{
& A = {\sin ^2}24 + \tan 34 + {\sin ^2}66 - \cot 56 - {{\cot 37} \over {\tan 53}} \cr
& = \left( {{{\sin }^2}24 + {{\sin }^2}66} \right) + \left( {\tan 34 - \cot 56} \right) - {{\cot 37} \over {\tan 53}} \cr
& {34^0} + {56^0} = {90^0} \Rightarrow \tan 34 = \cot 56 \Rightarrow \tan 34 - \cot 56 = 0 \cr
& {37^0} + {53^0} = {90^0} \Rightarrow \cot 37 = \tan 53 \Rightarrow {{\cot 37} \over {\tan 53}} = 1 \cr
& {\sin ^2}24 + {\sin ^2}66 = {{1 - \cos 48} \over 2} + {{1 - \cos 132} \over 2} = 1 - {{\cos 48 + \cos 132} \over 2} \cr
& {48^0} + {132^0} = {180^0} \Rightarrow \cos 48 = - \cos 132 \Rightarrow \cos 48 + \cos 132 = 0 \cr
& \Rightarrow {\sin ^2}24 + {\sin ^2}66 = 1 \cr
& Vay\,\,A = 1 + 0 - 1 = 0 \cr} $$
