Evaluate the following determinant: 15^ & 15^ 75^ & 75^ - Vidyadip

Question

Evaluate the following determinant:
$\begin{vmatrix}\cos15^\circ&\sin15^\circ\\\sin75^\circ&\cos75^\circ \end{vmatrix}$

✓

Answer

$\triangle=\cos15^\circ\cos75^\circ-\sin15^\circ\sin75^\circ$
$=\cos15^\circ\cos75^\circ-\sin(90^\circ-75^\circ)\sin(90^\circ-15^\circ)$ $[\because\sin(90^\circ-\theta)=\cos\theta]$
$=\cos15^\circ\cos75^\circ-\cos75^\circ\cos15^\circ$
$=\cos15^\circ\cos75^\circ-\cos15^\circ\cos75^\circ$
$=0$

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