Consider the following equality: a_1 \sin(x)+a_2 \sin^2(x)+\ldots+a_n \sin^n(x)=a_1 \sin(x)+

Lainey Goodwin

Lainey Goodwin

Answered question

2022-01-24

Consider the following equality:
a1sin(x)+a2sin2(x)++ansinn(x)=a1sin(x)+a2sin(2x)++ansin(nx)
where a1,a2,anC,nN
This equality clearly holds for n=1 and for any a1C. The trivial equality (a1=a2==an=0) also holds.
Do there exists any other solutions? If so, are there infinitely many of such solutions?
Do there exist other solutions if we constrain the domain of coefficients to Q, R or Z?

Answer & Explanation

Addisyn Thompson

Addisyn Thompson

Beginner2022-01-25Added 16 answers

With z=eix, you want to establish the identity
kak(zz12i)k=kakkz(kz)12i
By identification of the polynomial coefficients, only a1 can be nonzero.

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