Suppose I want to write the function $x \sin(t)$ as the series over the interval $x \in (0,\pi)$
$$x\sin(t) = \sum_{n=1}^{\infty}(a_n \cos(t) + b_n \sin(t) )\sin(nx)$$
Then would the coefficients $a_n$ and $b_n$ be just simply
$$a_n =\frac{1}{\pi \cos(t)} \int_{0}^{\pi}x\sin(nx) \;\mathrm{d}x = \frac{\tan(t)}{\pi} \int_{0}^{\pi}x\sin(nx) \;\mathrm{d}x $$
$$b_n =\frac{1}{\pi \sin(t)} \int_{0}^{\pi}x\sin(t)\sin(nx) \;\mathrm{d}x = \frac{1}{\pi} \int_{0}^{\pi}x\sin(nx) \;\mathrm{d}x$$
Could i just treat the trig functions as parameters?
Thanks
EDIT: The bounty is supposed to read "not enough attention". I may have misselcted the wrong item when I set the bounty
(Cf. my comment above)
The only sense one can give the formula $$x\sin(t) = \sum_{n=1}^{\infty}(a_n \cos(t) + b_n \sin(t) )\sin(x)$$ is the following: Assume that the (constant) series $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ both converge with sum $a$ and $b$ respectively. Then one would have the equation $$x\sin t=(a\cos t + b\sin t)\sin x$$ connecting the variables $t$ and $x$. Nothing about Fourier series here.
The formula $$a_n =\frac{1}{\pi \cos(t)} \int_{0}^{\pi}x\sin(x) \;\mathrm{d}x = \frac{\tan(t)}{\pi} \int_{0}^{\pi}x\sin(x) \;\mathrm{d}x $$ and the similar formula for $b_n$ doesn't make any sense whatsoever. Note that the letter $n$ isn't even appearing on the RHS.