Question

Evaluate the following determinants :
$\left|\begin{array}{cc}\cos {15}^{\circ }& \sin {15}^{\circ }\\ \sin {75}^{\circ }& \cos {75}^{\circ }\end{array}\right|$

Answer 1

$\left|\begin{array}{cc}cos{15}^{\circ }& sin{15}^{\circ }\\ sin{75}^{\circ }& cos{75}^{\circ }\end{array}\right|$

$=\left|\begin{array}{cc}cos{15}^{\circ }& sin{15}^{\circ }\\ sin({90}^{\circ }-{15}^{\circ })& cos({90}^{\circ }-{15}^{\circ })\end{array}\right|$

$=\left|\begin{array}{cc}cos{15}^{\circ }& sin{15}^{\circ }\\ cos{15}^{\circ }& sin{15}^{\circ }\end{array}\right|$

$=\cos {15}^{\circ }\sin {15}^{\circ }-\cos {15}^{\circ }\sin {15}^{\circ }$

$=0$