If A = sin²θ + cos⁴θ for any value of θ, then the value of A is?

Solution (By Examveda Team)

$$\eqalign{ & {\text{According to the question, }} \cr & {\text{A}} = {\sin ^2}\theta + {\text{co}}{{\text{s}}^4}\theta \cr & {\text{Put }}\theta = {90^ \circ }{\text{ for maximum value of A}} \cr & {\text{A}} = {\sin ^2}{90^ \circ } + {\text{co}}{{\text{s}}^4}{90^ \circ } \cr & {\text{A}} = 1 + 0 \cr & {\text{A}} = 1 \cr & {\text{Put }}\theta = {45^ \circ }{\text{ for minimum value of A}} \cr & {\text{A}} = {\sin ^2}{45^ \circ } + {\text{co}}{{\text{s}}^4}{45^ \circ } \cr & {\text{A}} = \frac{1}{2} + \frac{1}{4} \cr & {\text{A}} = \frac{3}{4} \cr & \therefore {\text{A lies in }}\frac{3}{4} \leqslant {\text{A}} \leqslant {\text{1}} \cr} $$