Question Number 206618 by MATHEMATICSAM last updated on 20/Apr/24
$$\mathrm{If}\:\mathrm{A}\:=\:\mathrm{sin}^{\mathrm{4}} \theta\:+\:\mathrm{cos}^{\mathrm{4}} \theta\:\mathrm{then}\:\mathrm{select}\:\mathrm{the}\: \\ $$$$\mathrm{correct}\:\mathrm{option}: \\ $$$$\left.\mathrm{i}\right)\:\mathrm{0}\:<\:\mathrm{A}\:<\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{ii}\right)\:\mathrm{1}\:<\:\mathrm{A}\:<\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\left.\mathrm{iii}\right)\:\frac{\mathrm{1}}{\mathrm{2}}\:\leq\:\mathrm{A}\:\leq\:\mathrm{1} \\ $$$$\left.\mathrm{iv}\right)\:\frac{\mathrm{3}}{\mathrm{2}}\:\leq\:\mathrm{A}\:\leq\:\mathrm{2} \\ $$
Answered by MM42 last updated on 20/Apr/24
$${A}=\left({sin}^{\mathrm{2}} \theta+{cos}^{\mathrm{2}} \theta\right)^{\mathrm{2}} −\mathrm{2}{sin}^{\mathrm{2}} \theta{cos}^{\mathrm{2}} \theta \\ $$$$=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}{sin}^{\mathrm{2}} \left(\mathrm{2}\theta\right)\Rightarrow{ans}:\:\:\frac{\mathrm{1}}{\mathrm{2}}\leqslant{A}\leqslant\mathrm{1}\:\:\:\checkmark \\ $$$$ \\ $$
Answered by Frix last updated on 20/Apr/24
$${A}=\frac{\mathrm{3}}{\mathrm{4}}+\frac{\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{4}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\leqslant{A}\leqslant\mathrm{1} \\ $$