Solve ${sin}^{6} 2 x + {cos}^{6} 2 x = 7 / 16$ .

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Solution

The given equation can be rewritten as.
$(1)^{3} - 3 {sin}^{2} 2 x {cos}^{2} 2 x (1) = 7 / 16$
or $3 {sin}^{2} 2 x {cos}^{2} 2 x = 9 / 16$
or $4 {sin}^{2} 2 x {cos}^{2} 2 x = 3 / 4$
or ${sin}^{2} 4 x = 3 / 4$
$∴ sin 4 x = \pm \frac{\sqrt{3}}{2} = \pm sin \frac{π}{3}$
$∴ x = \frac{n π}{4} \pm \frac{π}{12}$ .

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