Question

The value of the determinant
$\begin{vmatrix}cos^{2}\,54^{\circ}&cos^{2}\,36^{\circ }&cot\,135^{\circ }\\ sin^{2}\,53^{\circ }&cot\,135^{\circ }&sin^{2}\,37^{\circ }\\ cot\,135^{\circ }&cos^{2}\,25^{\circ }&cos^{2}\,65^{\circ }\end{vmatrix}$ is equal to

• A. -2
• B. -1
• C. 0
• D. 1

Answer 1

Answer: (C)

Let $\Delta = \begin{vmatrix}cos^{2}\,54^{\circ}&cos^{2}\,36^{\circ }&cot\,135^{\circ }\\ sin^{2}\,53^{\circ }&cot\,135^{\circ }&sin^{2}\,37^{\circ }\\ cot\,135^{\circ }&cos^{2}\,25^{\circ }&cos^{2}\,65^{\circ }\end{vmatrix}$
$= \begin{vmatrix}cos^{2}\,54^{\circ }&cos^{2}\left(90^{\circ} - 54^{\circ}\right)&-1\\ sin^{2}\left(90^{\circ } - 37^{\circ }\right)&-1&sin^{2}\,37^{\circ }\\ -1&cos^{2}\,25^{\circ }&cos^{2}\left(90^{\circ } - 25^{\circ }\right)\end{vmatrix}$
$ = \begin{vmatrix}cos^{2}\,54^{\circ }&sin^{2}\,54^{\circ }&-1\\ cos^{2}\,37^{\circ }&-1&sin^{2}\,37^{\circ }\\ -1&cos^{2}\,25^{\circ }&sin^{2}\,25^{\circ }\end{vmatrix}$
$C_{1} \to C_{1} + C_{2} + C_{3}$
$= \begin{vmatrix}cos^{2}\,54^{\circ }+sin^{2}\,54^{\circ }-1 &sin^{2}\,54^{\circ }&-1\\ cos^{2}\,37^{\circ }-1+sin^{2}\,37^{\circ }&-1&sin^{2}\,37^{\circ }\\ -1+cos^{2}\,25^{\circ }+sin^{2}\,25^{\circ }&cos^{2}\,25^{\circ }&sin^{2}\,25^{\circ }\end{vmatrix}$
$= \begin{vmatrix}0&sin^{2}\,54^{\circ }-1\\ 0&-1\,sin^{2}\,37^{\circ }\\ 0&cos^{2}\,25^{\circ }\,sin^{2}\,25^{\circ }\end{vmatrix} = 0$