Question

Let
$\Delta =\left|\begin{array}{ccc}{\sin }^2{35}^{\circ }& {\sin }^2{55}^{\circ }& \tan ({55}^{\circ }+{80}^{\circ })\\ {\sin }^2{11}^{\circ }& \tan ({79}^{\circ }+{56}^{\circ })& {\sin }^2{79}^{\circ }\\ {\cos }^2{18}^{\circ }& {\cos }^2{72}^{\circ }& \cot ({72}^{\circ }+{63}^{\circ })\end{array}\right|$
then $\Delta$ equals

• A. -1
• B. 0
• C. 2
• D. $(\sqrt{5}+1)/8$

Answer 1

The correct option is B 0

$\Delta =\left|\begin{array}{ccc}{\sin }^2{35}^{\circ }& {\sin }^2{55}^{\circ }& \tan ({55}^{\circ }+{80}^{\circ })\\ {\sin }^2{11}^{\circ }& \tan ({79}^{\circ }+{56}^{\circ })& {\sin }^2{79}^{\circ }\\ {\cos }^2{18}^{\circ }& {\cos }^2{72}^{\circ }& \cot ({72}^{\circ }+{63}^{\circ })\end{array}\right|$

$=\left|\begin{array}{ccc}{\sin }^2{35}^{\circ }& {\cos }^2{35}^{\circ }& \tan {135}^{\circ }\\ {\sin }^2{11}^{\circ }& \tan {135}^{\circ }& {\cos }^2{11}^{\circ }\\ {\cos }^2{18}^{\circ }& {\sin }^2{18}^{\circ }& \cot {135}^{\circ }\end{array}\right|$

$=\left|\begin{array}{ccc}{\sin }^2{35}^{\circ }& {\cos }^2{35}^{\circ }& -1\\ {\sin }^2{11}^{\circ }& -1& {\cos }^2{11}^{\circ }\\ {\cos }^2{18}^{\circ }& {\sin }^2{18}^{\circ }& -1\end{array}\right|$

applying $C_1\to C_1+C_2+C_3$ gives

$=\left|\begin{array}{ccc}0& {\cos }^2{35}^{\circ }& -1\\ 0& -1& {\cos }^2{11}^{\circ }\\ 0& {\sin }^2{18}^{\circ }& -1\end{array}\right|=0$

$\therefore \Delta =0$