If f k ( x ) = 1 k ( sin k x + cos k x ) , where x R , k 1 , then f 4 ( x ) - f 6 ( x ) is equal to

Question

If $f_{k} (x) = \frac{1}{k} (\sin^{k} x + \cos^{k} x),$ where $x \in R, k \geq 1$ , then $f_{4} (x) - f_{6} (x)$ is equal to

Smart Achievers · Accepted Answer

(4)

Given, $f_{k} (x) = \frac{1}{k} (\sin^{k} x + \cos^{k} x),$ where $x \in R$ and $k \geq 1$

Now, $f_{4} (x) - f_{6} (x)$

$= \frac{1}{4} (\sin^{4} x + \cos^{4} x) - \frac{1}{6} (\sin^{6} x + \cos^{6} x)$

$= \frac{1}{4} (1 - 2 \sin^{2} x \cdot \cos^{2} x) - \frac{1}{6} (1 - 3 \sin^{2} x \cdot \cos^{2} x)$

$= \frac{1}{4} - \frac{1}{6} = \frac{1}{12}$

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