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  4. Let fk (x) = (1/k) ( sink x + cosk x) for k = 1, 2, 3, .... Then for all x ∈ R, the value of f4(x) - f6(x) is equal to :

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Q. Let $f_k (x) = \frac{1}{k} (\sin^k x + \cos^k x) $ for $k = 1, 2, 3, ....$ Then for all x $\in$ R, the value of $f_4(x) - f_6(x)$ is equal to :

JEE MainJEE Main 2019Trigonometric Functions

Solution:

$f_{4} \left(x\right) -f_{6}\left(x\right)$
$ =\frac{1}{4}\left(\sin^{4} x+\cos^{4}x\right)- \frac{1}{6} \left(\sin^{6}x+\cos^{6}x\right) $
$= \frac{1}{4} \left(1- \frac{1}{2} \sin^{2} 2x\right) - \frac{1}{6} \left(1- \frac{3}{4} \sin^{2} 2x \right) = \frac{1}{12} $