Adequacy: ▲c 1 = C2 =...=cn=0 becomes c1* sinx+C2 * sin2x+...+cn * sinnx = 0, and the sufficiency is proved.

Necessity: c1* sinx+C2 * sin2x+...+cn * sinnx is regarded as the product of two n-dimensional vectors.

Let a=(c 1, c2, ... cn) and b=(sinx, sin2x, ..., sinnx).

Then ab = c 1 * sinx+C2 * sin2x+...+cn * sinnx = lalblcos α, because lbl and α both change with the change of X. To make this formula constant, only lal is zero, and c1? +c2？ +......+cn？ =0, that is, c 1=c2=...=cn=0. So the necessity of obtaining a certificate is established.

To sum up, it can be concluded that for any real number X, there is C 1 * sinx+C2 * sin2x+...+CN * sinnx = 0 (n is a natural number, c1,C2, ... cn is a real number) if and only if c1= C2 =