Question

Suppose $sin^3 x sin 3x=\displaystyle\sum _{m=0}^{n}C_m$ cos nx is an identity in x, $where C_0, C_1, ...., C_n $ are constants and $C_n\ne 0.$ value of n is.......

Answer 1

Given, $sin^3 x sin 3x=\displaystyle\sum _{m=0}^{n}C_m$ cos nx is an identity in x
where, $C_0, C_1, ...., C_n constants.$
$sin^3 x sin 3x=\frac{1}{4}\{3 sin x-sin 3x\}.sin 3x$
$=\frac{1}{4}\Bigg(\frac{3}{2}.2 sin x.sin 3x-sin^2 3x\Bigg)$
$=\frac{1}{4}\Bigg\{\frac{3}{2}(cso 2x -cosx)-\frac{1}{2}(1-cos 6x)\Bigg\}$
$=\frac{1}{8}(cos 6x +3 cos 2x=3 cos x-1)$
$\therefore $ On comparing both sides, we get n = 6