The equation ${sin}^{4} x + {cos}^{4} x = a$ has a real solution, if

0 < a \leq 1

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\frac{1}{2} \leq a \leq 1

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\frac{1}{4} \leq a \leq \frac{1}{2}

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- 1 \leq a \leq 1

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Solution

The correct option is C $\frac{1}{2} \leq a \leq 1$
${sin}^{4} x + {cos}^{4} x = a$
$\Rightarrow {(\frac{1 - cos 2 x}{2})}^{2} + {(\frac{1 + cos 2 x}{2})}^{2} = \frac{1 + {cos}^{2} 2 x}{2} = a$
$\Rightarrow \frac{1}{2} \leq a \leq 1$ since, $0 \leq {cos}^{2} 2 x \leq 1$

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