If ${\Delta _1} = \left| {\begin{array}{*{20}{c}} x & {\sin \theta } & {\cos \theta } \\ {\sin \theta } & { - x} & 1 \\ {\cos \theta } & 1 & x \end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}} x & {\sin 2\theta } & {\cos 2\theta } \\ {\sin 2\theta } & { - x} & 1 \\ {\cos 2\theta } & 1 & x \end{array}} \right|$,$x \ne 0$; then for all $\theta \in \left( {0, \frac{\pi }{2}} \right)$:

• A
  ${\Delta _1} - {\Delta _2} = - 2{x^3}$
• B
  ${\Delta _1} + {\Delta _2} = - 2({x^3} + x - 1)$
• C
  ${\Delta _1} - {\Delta _2} = x(\cos 2\theta - \cos 4\theta )$
• D
  ${\Delta _1} + {\Delta _2} = - 2{x^3}$