91Ó°ÊÓ

Find all values of \(\theta\) lying between 0 and \(\frac{\pi}{2}\) which satisfy the equation $$ \left|\begin{array}{ccc} 1+\sin ^{2} \theta & \cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & 1+\cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & \cos ^{2} \theta & 1+4 \sin 4 \theta \end{array}\right|=0 $$

Using determinants, find the area of the triangle whose vertices are \((1,4),(2,3)\) and \((-5,-3)\) Are the given points collinear?