Let`f_(4)(x) = (1)/(k) [ sin^(k) + cos^(k) x ] `where x `in RR` and `k ge`1.
then `f_(4)(x) - f_(6)(x)` =

Options

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

Answer

A

Explanation

We have,
`f_(k)(x) = (1)/(k)(sin^(h)x+ cos^(k)x), k = 1, 2, 3, ...`
`therefore f_(4)(x) = (1)/(4)(sin^(4)x + cos^(4) x)`
`=(1)/(4)((sin^(2)x + cos^(2)x)^(2)-2sin^(2)xcos^(2)x)`
`=(1)/(4)(1-(1)/(2)(sin2x)^(2))=(1)/(4)-(1)/(8)sin^(2)2x`
and `f_(6)(x)=(1)/(6)(sin^(6)x + cos^(6)x)`
`=(1)/(6){(sin^(2)x+cos^(2)x)^(3)-3sin^(2) x cos^(2)x (sin^(2)x + cos^(2)x)}`
`=(1)/(6){1-(3)/(4)(2sin x cos x)^(2)}=(1)/(6)-(1)/(8)sin^(2)2x`
Now, `f_(4)(x) - f_(6)(x)=(1)/(4)-(1)/(6)=(3-2)/(12)=(1)/(12)`