Given trig polynomials $$T(x)= \sum_{k=0}^{n} a_k\cos(kx)+b_k\sin(kx) $$ $$V(x)= \sum_{k=0}^{l} \alpha_k\cos(kx)+\beta_k\sin(kx)$$
I want to show that the product $T(x)V(x)$ is also a trig polynomial. I know $A=\{1,\cos(x), \cos(2x),\ldots,\cos(nx),\sin(x),\sin(2x),\ldots,\sin(nx)\}$ is an orthonormal basis for the space of all degree $n$ trig polynomials with the inner product $\langle f,g \rangle = \int_{-\pi}^{\pi} f(x)g(x)\ dx $. I've been trying to show that $\langle TV,f \rangle =0$ for some $f\in A$, but I have had much luck. Any suggestions?
Hint
$$\sin(kx)\sin(mx)=\frac{1}{2} [ \cos((k-m)x)- \cos((k+m)x)]$$ $$\cos(kx)\cos(mx)=\frac{1}{2} [ \cos((k-m)x)+ \cos((k+m)x)]$$ $$\sin(kx)\cos(mx)=\frac{1}{2} [ \sin((k+m)x)+ \sin((k-m)x)]$$