Knowing that $$\left\{ \sin\left(kx\right)\right\} _{k\in\mathbb{N}}$$ and $$\left\{ \cos\left(kx\right)\right\} _{k\in\mathbb{N\cup}\left\{ 0\right\} }$$
are complete systems in $L^2(0,\pi)$. How would we prove that $$\left\{ \sin\left(\left(k+\frac{1}{2}\right)x\right)\right\} _{k\in\mathbb{N}}$$ and $$\left\{ \cos\left(\left(k+\frac{1}{2}\right)x\right)\right\} _{k\in\mathbb{N\cup}\left\{ 0\right\} }$$ are also complete systems that generate any function $f\in{L^2(0,\pi)}$?
I have been using this to solve PDE's with robin conditions. It works, but I don't see any justification why this can be done.
You would need to show that
$$\int_0^{\pi} dx \sin{\left [ \left (k+\frac{1}{2} \right)x \right]} \sin{\left [ \left (k'+\frac{1}{2} \right)x \right]} = \begin{cases} \\ 0 & k \ne k'\\ M_k & k = k'\end{cases}$$
$$\int_0^{\pi} dx \sin{\left [ \left (k+\frac{1}{2} \right)x \right]} \cos{\left [ \left (k'+\frac{1}{2} \right) x \right]} = \begin{cases} \\ 0 & k \ne k'\\ N_k & k = k'\end{cases}$$
$$\int_0^{\pi} dx \cos{\left [ \left (k+\frac{1}{2} \right)x \right]} \cos{\left [ \left (k'+\frac{1}{2} \right)x \right]} = \begin{cases} \\ 0 & k \ne k'\\ P_k & k = k'\end{cases}$$
where $M_k$, $N_k$, and $P_k$ are nonzero normalization factors.