Find the fundamental period for $f (x) = {sin}^{4} x + {cos}^{4} x$

\frac{π}{4}

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\frac{π}{2}

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π

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2 π

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Solution

The correct option is B $\frac{π}{2}$
$f (x) = {sin}^{4} x + {cos}^{4} x$
$f (x) = ({sin}^{2} x + {cos}^{2} x)^{2} - 2 {sin}^{2} x {cos}^{2} x$
$f (x) = 1 - \frac{1}{2} (4 {sin}^{2} x {cos}^{2} x)$
$f (x) = 1 - \frac{1}{2} {sin}^{2} 2 x$
$f (x) = 1 - \frac{1}{4} (1 - cos 4 x)$
$\Rightarrow f (x) = \frac{3}{4} + \frac{1}{4} cos 4 x$
Now, period of $cos 4 x$ is $\frac{2 π}{4} = \frac{π}{2}$
Hence, fundamental period of $f (x)$ is $\frac{π}{2}$

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