Prove $\sqrt2\sin(\pi pj\Delta x)$ is ONB of $\{U \in \mathbb{R}^{M+1} \mid U_1=U_M=0\}$ w.r.t. $(U,W)=\Delta x\sum_{j=0}^{M}U_jW_j$

Question

Here is the full exercise:

From my understanding I need to show:

• $||\varphi_p||^2=\frac{1}{\Delta x}\sum_{j=0}^{M}\sin^2(\pi pj\Delta x)=1 $
• $p \neq q \implies (\varphi_p,\varphi_q)=0$
• $\operatorname{span}(\{\varphi_p\}_{p=1}^{M-1}) = \{U \in \mathbb{R}^{M+1} \mid U_1=U_M=0\}$

for the first part. I managed to show the unit length with trig identities, exponential notation and geometric series. But for step 2 I am at a loss as to how to rigorously prove, beyond a handwavy symmetry argument, that the cos terms you get from rewriting $sin \alpha sin \beta$ as $cos(\alpha+\beta)-cos(\alpha-\beta)$ cancel to produce a scalar product of $0$.

I solved the span by first truncating $\varphi_p$ and $U\in\mathbb{R}^{M+1}$ ("deleting" their 1st and last component, because those are constantly zero anyway) and showing that the truncated vectors are still orthonormal, beyond that their matrix representation is symmetric, thus orthonormal, i.e. invertible.

Any hint regarding the second part of the exercise, i.e. eigenfunctions of the centered difference operator, would be appreciated.