The value of $$\frac{{{{\sec }^2}\theta }}{{{\text{cose}}{{\text{c}}^2}\theta }} + \frac{{{\text{cose}}{{\text{c}}^2}\theta }}{{{{\sec }^2}\theta }} - \left( {{{\sec }^2}\theta + {\text{cose}}{{\text{c}}^2}\theta } \right){\text{is:}}$$

$$\eqalign{ & \frac{{{{\sec }^2}\theta }}{{{\text{cose}}{{\text{c}}^2}\theta }} + \frac{{{\text{cose}}{{\text{c}}^2}\theta }}{{{{\sec }^2}\theta }} - \left( {{{\sec }^2}\theta + {\text{cose}}{{\text{c}}^2}\theta } \right) \cr & = \frac{{{{\sec }^4}\theta + {{\cos }^4}\theta }}{{{{\sin }^2}\theta .{{\cos }^2}\theta }} - \left( {\frac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\sin }^2}\theta .{{\cos }^2}\theta }}} \right) \cr & = \frac{{\left( {{{\sin }^4}\theta - {{\sin }^2}\theta } \right) + \left( {{{\cos }^4}\theta - {{\cos }^2}\theta } \right)}}{{{{\sin }^2}\theta .{{\cos }^2}\theta }} \cr & = \frac{{ - {{\sin }^2}\theta .{{\cos }^2}\theta - {{\sin }^2}\theta .{{\cos }^2}\theta }}{{{{\sin }^2}\theta .{{\cos }^2}\theta }} \cr & = - 2 \cr & {\bf{Alternate:}} \cr & {\text{Put }}\theta = {45^ \circ } \cr & 1 + 1 - \left( {2 + 2} \right) = - 2 \cr} $$