Prove that $\sin(x+a), \sin(x+b),\sin(x+c), \hspace{5pt} a,b,c \in \mathbb{R}$ are linearly dependent I just want an answer verification (or not).
We have that:
$\begin{array}[t]{l}
\sin(x+a)=\cos a\cdot \sin x + \sin a \cdot \cos x\\
\sin(x+b)=\cos b\cdot \sin x + \sin b \cdot \cos x\\
\sin(x+c)=\cos c\cdot \sin x + \sin c \cdot \cos x\\
\end{array}$
Since $\sin(x+a), \sin (x+b), \sin(x+c) \in  \hspace{5pt} V=\langle\cos x , \sin x\rangle$ and $\dim V=2\implies$ given functions cannot be a basis of $V$. Consequently, they are linearly dependent.

Your solution is correct. $${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$$
