Determinant of matrix with trigonometric functions Find the determinant of the following matrix:
  $$\begin{pmatrix}\cos\left(a_{1}-b_{1}\right) &amp; \cos\left(a_{1}-b_{2}\right) &amp; \cos\left(a_{1}-b_{3}\right)\\
\cos\left(a_{2}-b_{1}\right) &amp; \cos\left(a_{2}-b_{2}\right) &amp; \cos\left(a_{2}-b_{3}\right)\\
\cos\left(a_{3}-b_{1}\right) &amp; \cos\left(a_{3}-b_{2}\right) &amp; \cos\left(a_{3}-b_{3}\right)
\end{pmatrix}$$For $a_1,\dots ,a_3,b_1,\dots, b_3\in\mathbb{R}$


I'm completely stumped honestly. I tried using the cosine addition identity to open the cosine, but I wasn't able to find how it helps me, and even for a $2\times2$ version of the matrix I wasn't really sure what to do. Any help?
The matrix is the product
\begin{eqnarray}
\begin{pmatrix}\cos a_1&amp; \sin a_1&amp; 0\\ \cos a_2&amp;\sin a_2&amp; 0\\ \cos a_3&amp;\sin a_3 &amp;0\end{pmatrix}\begin{pmatrix}\cos b_1&amp; \cos b_2&amp;\cos b_3\\ \sin b_1&amp;\sin b_2&amp;\sin b_3\\ 0&amp;0&amp;0\end{pmatrix}
\end{eqnarray}
and thus the determinant is 0.
Find the determinant of the following trigonometric matrix(unit in radian):

sin1 sin2 sin3 sin4 sin5                                                              

sin2 sin3 sin4 sin5 sin6

sin3 sin4 sin5 sin6 sin7

sin4 sin5 sin6 sin7 sin8

sin5 sin6 sin7 sin8 sin9