Question

Suppose $si{n}^{3}xsin3x=\sum _{m=0}^n{C}_m$ 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, $si{n}^{3}xsin3x=\sum _{m=0}^n{C}_m$ cos nx is an identity in x
where, $C_0,C_1,....,C_{n}constants.$
$si{n}^{3}xsin3x=\frac{1}{4}\{3sinx-sin3x\}.sin3x$
$=\frac{1}{4}(\frac{3}{2}.2sinx.sin3x-si{n}^{2}3x)$
$=\frac{1}{4}\{\frac{3}{2}(cso2x-cosx)-\frac{1}{2}(1-cos6x)\}$
$=\frac{1}{8}(cos6x+3cos2x=3cosx-1)$
$\therefore$ On comparing both sides, we get n = 6