Let `F_(k)(x)=1/k (sin^(k)x+cos^(k)x)`, where `x in R` and `k ge 1`, then find the value of `F_(4)(x)-F_(6)(x)`.

Options

A. `1/4`
B. `1/12`
C. `1/6`
D. `1/3`

Answer

B

Explanation

We have, `f_(k)(x)=1/k(sin^(k)x+cos^(k)x)`
`thereforef_(4)(x)-f_(6)(x)`
`=1/4(sin^(4)x+cos^(4)x)-1/6(sin^(6)x+cos^(6)x)`
`=1/4{(sin^(2)x+cos^(2)x)^(2)-2sin^(2)xcos^(2)x}`
`-1/6{(sin^(2)x+cos^(2)x)^(3)-3sin^(2)x cos^(2)(sin^(2)x+cos^(2)x)}`
`=1/4(1-2sin^(2)xcos^(2)x)-1/6(1-3sin^(2)xcos^(2)x)=1/12`