If `sin x_(1)+sin x_(2) + sin x_(3)+...+sin x_(2008)=2008` then find the value of `sin^(2008) x_( 1)+sin^(2008)x_(2) + sin^(2008)x_(3)+...+sin^(2008)x_(2008)`.

Explanation

We know that
`sinx_(i) le 1 AA i`
`rArr sinx_(1) + sinx_(2) + sinx_(3) +……….+ sinx_(2009) le 2008`.
Thus, equality holds only when each of the terms is 1.
i.e., `sinx_(i) =1 AA i=1,2,3,…………….,2008`
and consequently `cos x_(i)=0, AAi=1,2,3,............,2008`
Now, `sin^(2008)x_(1) + sin^(2008)x_(2) + sin^(2008)x_(3) + ..........+sin^(2008)x_(2008) = 1+1+1+............. +1=2008`
and`cos^(2009)x_(1) + cos^(2009)x_(2) + cos^(2009)x_(3) +......+ cos^(2009)x_(2008)=0+0+0+.................)0=0`