The value of the expression (cos⁶θ + sin⁶θ - 1)(tan²θ + cot²θ + 2) is:

$$\eqalign{ & \left( {{{\cos }^6}\theta + {{\sin }^6}\theta - 1} \right)\left( {{{\tan }^2}\theta + {{\cot }^2}\theta + 2} \right) \cr & {\text{Put }}\theta = {45^ \circ } \cr & = \left( {{{\cos }^6}{{45}^ \circ } + {{\sin }^6}{{45}^ \circ } - 1} \right)\left( {{{\tan }^2}{{45}^ \circ } + {{\cot }^2}{{45}^ \circ } + 2} \right) \cr & = \left( {\frac{1}{8} + \frac{1}{8} - 1} \right)\left( {1 + 1 + 2} \right) \cr & = \frac{{ - 3}}{4} \times 4 \cr & = - 3 \cr} $$