$A = {\sin ^2}x + {\cos ^4}x$ $= {\sin ^2}x + {\cos ^2}x\left( {1 - {{\sin }^2}x} \right)$ $= {\sin ^2}x + {\cos ^2}x - {1 \over 4}{\left( {2\sin x.\cos x} \right)^2}$ $= 1 - {1 \over 4}{\sin ^2}\left( {2x} \right)$ Now $0 \le {\sin ^2}\left( {2x} \right) \le 1$ $\Rightarrow 0 \ge - {1 \over 4}{\sin ^2}\left( {2x} \right) \ge - {1 \over 4}$ $\Rightarrow 1 \ge 1 - {1 \over 4}{\sin ^2}\left( {2x} \right) \ge 1 - {1 \over 4}$ $\Rightarrow 1 \ge A \ge {3 \over 4}$