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 :

Question

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 : (A) (5/12) (B) (-1/12) (C) (1/4) (D) (1/12)

Tardigrade Admin · Accepted Answer

f4 (x) -f6(x) =(1/4)( sin4 x+ cos4x)- (1/6) ( sin6x+ cos6x) = (1/4) (1- (1/2) sin2 2x) - (1/6) (1- (3/4) sin2 2x ) = (1/12)

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 :

$\frac{5}{12}$

$\frac{- 1}{12}$

$\frac{1}{4}$

$\frac{1}{12}$

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

Q. Let fk​(x)=k1​(sinkx+coskx) for k=1,2,3,.... Then for all x ∈ R, the value of f4​(x)−f6​(x) is equal to :

125​

12−1​

41​

121​