Question

The value of the determinant
$\left|\begin{array}{ccc}{\sin }^{2}36^{\circ }& {\cos }^{2}36^{\circ }& \cot 135^{\circ }\\ {\sin }^{2}53^{\circ }& \cot 135^{\circ }& {\cos }^{2}53^{\circ }\\ \cot 135^{\circ }& {\cos }^{2}25^{\circ }& {\cos }^{2}65^{\circ }\end{array}\right|$ is

• A. - 2
• B. - 1
• C. 0
• D. 1
• E. 2

Answer 1

Answer: (C)

$\Delta =\left|\begin{array}{ccc}{\sin }^{2}36^{\circ }& {\cos }^{2}36^{\circ }& \cot 135^{\circ }\\ {\sin }^{2}53^{\circ }& \cot 135^{\circ }& {\cos }^{2}53^{\circ }\\ \cot 135^{\circ }& {\cos }^{2}25^{\circ }& {\cos }^{2}65^{\circ }\end{array}\right|$
$\therefore {\cos }^{2}36^{\circ }={\cos }^2(90-54)={\sin }^{2}54^{\circ }$
${\sin }^{2}37^{\circ }={\sin }^2(90-53{)}^{\circ }={\cos }^{2}53^{\circ }$
${\cos }^{2}65^{\circ }={\cos }^2(90-25{)}^{\circ }={\sin }^{2}25^{\circ }$
$\cot 135^{\circ }=\cot (90+45{)}^{\circ }=-\tan 45^{\circ }=-1$
$\therefore \Delta =\left|\begin{array}{ccc}{\cos }^{2}54^{\circ }& {\sin }^{2}54^{\circ }& -1\\ {\sin }^{2}53& -1& {\cos }^{2}53^{\circ }\\ -1& {\cos }^{2}25^{\circ }& {\sin }^{2}25^{\circ }\end{array}\right|$
$\Delta =\left|\begin{array}{c}{\cos }^{2}54^{\circ }+{\sin }^{2}54^{\circ }-1{\sin }^{2}54^{\circ }-1\\ {\sin }^{2}53^{\circ }-1+{\cos }^{2}53^{\circ }-1co{s}^{2}53^{\circ }\\ -1+{\cos }^{2}25^{\circ }+{\sin }^{2}25^{\circ }{\cos }^{2}25^{\circ }{\sin }^{2}25^{\circ }\end{array}\right|$
$=\left|\begin{array}{ccc}0& {\sin }^{2}54^{\circ }& -1\\ 0& -1& {\cos }^{2}53^{\circ }\\ 0& {\cos }^{2}25^{\circ }& {\sin }^{2}25^{\circ }\end{array}\right|=0$