I have two known time series $ Y, $
$Y = \left\{ Y_i \right\}_N \hspace {0.2 in} Y_i > 0 \hspace {0.2 in} -1 \leq Y_{i+1} - Y_i \leq 1 $ for all $i$
$ = \left\{ _i \right\}_N $
Let us define the time series $φ$ as:
\begin{equation} y_j = Y_{j+1} - Y_j \hspace {0.2 in} y_j = cos(φ_j + _j) \hspace {0.2 in} φ = \left\{ φ_i \right\}_N \hspace {0.2 in} \end{equation}
then \begin{equation} Y_{i+1} - Y_0 = \sum_{j=0}^{j=i}cos(φ_j + _j) \end{equation}
and \begin{equation} Y_{i+1} - Y_0 = \sum_{j=0}^{j=i}cos(φ_j)cos(_j) - \sum_{j=0}^{j=i}sin(φ_j)sin( _j) \end{equation}
After plotting I found that the charts of
\begin{equation} \sum_{j=0}^{j=i}cos(φ_j) \hspace {0.2 in} \sum_{j=0}^{j=i}sin(φ_j) \end{equation}
are the same respectively to \begin{equation} \sum_{j=0}^{j=i}sin(_j) \hspace {0.2 in} \sum_{j=0}^{j=i}cos(_j) \end{equation}
How is it possible?