The value of $\left| {\begin{array}{*{20}{c}}
{\sin \alpha }&{\cos \alpha }&{\sin \left( {\alpha  + \gamma } \right)}\\
{\sin \beta }&{\cos \beta }&{\sin \left( {\beta  + \gamma } \right)}\\
{\sin \delta }&{\cos \delta }&{\sin \left( {\gamma  + \delta } \right)}
\end{array}} \right|$ is 

If $\omega $ is an imaginary root of unity, then the value of $\left| {\,\begin{array}{*{20}{c}}a&{b{\omega ^2}}&{a\omega }\\{b\omega }&c&{b{\omega ^2}}\\{c{\omega ^2}}&{a\omega }&c\end{array}\,} \right|$ is

The values of the determinant $\left| {\,\begin{array}{*{20}{c}}1&{\cos (\alpha - \beta )}&{\cos \alpha }\\{\cos (\alpha - \beta )}&1&{\cos \beta }\\{\cos \alpha }&{\cos \beta }&1\end{array}\,} \right|$ is

If $\Delta = \left| {\,\begin{array}{*{20}{c}}x&y&z\\p&q&r\\a&b&c\end{array}\,} \right|,$ then $\left| {\,\begin{array}{*{20}{c}}x&{2y}&z\\{2p}&{4q}&{2r}\\a&{2b}&c\end{array}\,} \right|$equals

$\left| {\,\begin{array}{*{20}{c}}1&5&\pi \\{{{\log }_e}e}&5&{\sqrt 5 }\\{{{\log }_{10}}10}&5&e\end{array}\,} \right| = $

The system of equations  $-k x+3 y-14 z=25$  $-15 x+4 y-k z=3$  $-4 x+y+3 z=4$  is consistent for all $k$ in the set