Question

The value of the determinant
$\left|\begin{array}{ccc}co{s}^{2}54^{\circ }& co{s}^{2}36^{\circ }& cot135^{\circ }\\ si{n}^{2}53^{\circ }& cot135^{\circ }& si{n}^{2}37^{\circ }\\ cot135^{\circ }& co{s}^{2}25^{\circ }& co{s}^{2}65^{\circ }\end{array}\right|$ is equal to

• A. -2
• B. -1
• C. 0
• D. 1

Answer 1

Answer: (C)

Let $\Delta =\left|\begin{array}{ccc}co{s}^{2}54^{\circ }& co{s}^{2}36^{\circ }& cot135^{\circ }\\ si{n}^{2}53^{\circ }& cot135^{\circ }& si{n}^{2}37^{\circ }\\ cot135^{\circ }& co{s}^{2}25^{\circ }& co{s}^{2}65^{\circ }\end{array}\right|$
$=\left|\begin{array}{ccc}co{s}^{2}54^{\circ }& co{s}^2\left(90^{\circ }-54^{\circ }\right)& -1\\ si{n}^2\left(90^{\circ }-37^{\circ }\right)& -1& si{n}^{2}37^{\circ }\\ -1& co{s}^{2}25^{\circ }& co{s}^2\left(90^{\circ }-25^{\circ }\right)\end{array}\right|$
$=\left|\begin{array}{ccc}co{s}^{2}54^{\circ }& si{n}^{2}54^{\circ }& -1\\ co{s}^{2}37^{\circ }& -1& si{n}^{2}37^{\circ }\\ -1& co{s}^{2}25^{\circ }& si{n}^{2}25^{\circ }\end{array}\right|$
$C_1\to C_1+C_2+C_3$
$=\left|\begin{array}{ccc}co{s}^{2}54^{\circ }+si{n}^{2}54^{\circ }-1& si{n}^{2}54^{\circ }& -1\\ co{s}^{2}37^{\circ }-1+si{n}^{2}37^{\circ }& -1& si{n}^{2}37^{\circ }\\ -1+co{s}^{2}25^{\circ }+si{n}^{2}25^{\circ }& co{s}^{2}25^{\circ }& si{n}^{2}25^{\circ }\end{array}\right|$
$=\left|\begin{array}{cc}0& si{n}^{2}54^{\circ }-1\\ 0& -1si{n}^{2}37^{\circ }\\ 0& co{s}^{2}25^{\circ }si{n}^{2}25^{\circ }\end{array}\right|=0$