<s>[INST] Prove the following determinant identities without expanding the determinants a)
$$\begin{vmatrix}
  \sin^2 x & \cos^2 x & \cos 2x \\
  \sin^2 y & \cos^2 y & \cos 2y \\
  \sin^2 z & \cos^2 z & \cos 2z \\
\end{vmatrix} = 0;$$
$$\begin{vmatrix}
  \sin^2 x & \cos^2 x & \cos^2x-\sin^2x \\
  \sin^2 y & \cos^2 y & \cos^2y-\sin^2y \\
  \sin^2 z & \cos^2 z & \cos^2z-\sin^2z \\
\end{vmatrix} = 0;$$
$$\begin{vmatrix}
  \cos^2 x & \cos^2 x & \cos^2x-\sin^2x \\
  \cos^2 y & \cos^2 y & \cos^2y-\sin^2y \\
  \cos^2 z & \cos^2 z & \cos^2z-\sin^2z \\
\end{vmatrix} = 0;$$
b)
$$\begin{vmatrix}
   1 & a & p+c \\
   1 & p & c+a \\
   1 & c & a+p \\
\end{vmatrix} = 0,$$
$$\begin{vmatrix}
    1 & a & 1\cdot(a+p+c) \\
    1 & p & 1\cdot(p+c+a) \\
    1 & c & 1\cdot(c+a+p)
\end{vmatrix} =
(a + p + c)\cdot
\begin{vmatrix}
    1 & a & 1 \\
    1 & p & 1 \\
    1 & c & 1
\end{vmatrix} = 0.$$
I am stuck on a), any hints and help is appreciate, and please check if b) is correct.
 [/INST] Remember that $\cos 2x = \cos^2 x - \sin^2 x$.
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