Solve $s i n^{4} (x / 3) + c o s^{4} (x / 3) > 1 / 2$ .

R - {x : x = \frac{3 n π}{2} \pm \frac{3 π}{4}, n ϵ z}

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R - {x : x = n π \pm \frac{3 π}{4}, n ϵ z}

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R - {x : x = 2 n π \pm \frac{3 π}{4}, n ϵ z}

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R - {x : x = \frac{n π}{2} \pm \frac{3 π}{4}, n ϵ z}

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Solution

The correct option is A $R - {x : x = \frac{3 n π}{2} \pm \frac{3 π}{4}, n ϵ z}$
${sin}^{4} (\frac{x}{3}) + {cos}^{4} (\frac{x}{3}) > \frac{1}{2}$
$\Rightarrow 1 + 2 {sin}^{2} (\frac{x}{3}) {cos}^{2} (\frac{x}{3}) > \frac{1}{2}$
$\Rightarrow \frac{1}{2} {sin}^{2} (\frac{2 x}{3}) > - \frac{1}{2}$
$\Rightarrow {sin}^{2} (\frac{2 x}{3}) > - 1$
$\Rightarrow R - (x : x = \frac{3 n π}{2} \pm \frac{3 π}{4})$

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