Polynomial division question with trig and bynomial

Question

I have the polynomial $f=(\cos a+x\sin a)^n$ and the polynomial $g=x^2+1$ and I have to find the remainder of the divison: $\frac fg$.

I tried something using the euclidean division but couldn't work it out..

Answer 1

Since $g(x)$ has degree $2$, the degree of the remainder is $1$, at most. That is, the remainder is $bx+c$, for some $b,c\in\mathbb C$.

Let $q(x)$ be the quotient. Then $f(x)=q(x)(x^2+1)+bx+c$ and therefore$$\left\{\begin{array}{l}f(i)=bi+c\\f(-i)=-bi+c.\end{array}\right.$$But $f(i)=e^{ian}$ and $f(-i)=e^{-ian}$. So$$\left\{\begin{array}{l}bi+c=e^{ian}\\-bi+c=e^{-ian}.\end{array}\right.$$Therefore, $b=\sin(na)$ and $c=\cos(na)$.