If `8sin^(3)xsin3xsin3x=sum_(r=0)^(n)a^(r)cosrx` is an identity in x, then n=

Options

A. 3
B. 4
C. 6
D. 9

Answer

C

Explanation

`8sin^(3)xsin3x=2sin3x(3sinx-sin3x)`
`implies8sin^(3)x sin3x=6sin3xsinx-2 sin^(2)3x`
`implies8sin^(3)x sin3x=3(cos2x-cos4x)-(-cos6x)`
`implies8sin^(3)x sin3x=-1+3cos2x-3 cos4x+cos6x`
`therefore8sin^(3)x sin3x=underset(r=0)overset(n)(sum)a^(r)cos rx`
`implies-1+3cos2x-3cos4x+cos6x`
`" "=a_(0)+a_(1)cosx+...+a_(n)cosnx`
`impliesn=6.`