Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

De determinant $https://latex.codecogs.com/png.image?\large \dpi{110}\begin{vmatrix}\sin a & 1\!+\!\cos 2a & \sin^2 a \\\sin b & 1\!+\!\cos 2b & \sin^2 b \\\sin c & 1\!+\!\cos 2c & \sin^2 c \\\end{vmatrix}$ is gelijk aan	A. (sin a − sin b)(sin b − sin c)(sin c − sin a)
B. 2 (sin a − sin b)(sin b − sin c)(sin c − sin a)
C. − 2 (sin a − sin b)(sin b − sin c)(sin c − sin a)
D. − (sin a − sin b)(sin b − sin c)(sin c − sin a)
E. 0 (nul)

The determinant $https://latex.codecogs.com/png.image?\large \dpi{110}\begin{vmatrix}\sin a & 1\!+\!\cos 2a & \sin^2 a \\\sin b & 1\!+\!\cos 2b & \sin^2 b \\\sin c & 1\!+\!\cos 2c & \sin^2 c \\\end{vmatrix}$ is equal to	A. (sin a − sin b)(sin b − sin c)(sin c − sin a)
	B. 2 (sin a − sin b)(sin b − sin c)(sin c − sin a)
	C. − 2 (sin a − sin b)(sin b − sin c)(sin c − sin a)
	D. − (sin a − sin b)(sin b − sin c)(sin c − sin a)
	E. 0 (nul)

$\\\begin{vmatrix}\sin a & 1+\cos 2a & \sin^2 a \\\sin b & 1 +\cos2b & \sin^2 b \\\sin c & 1+\cos 2c & \sin^2 c \\\end{vmatrix}=\begin{vmatrix}\sin a & 2.\cos^2a & \sin^2 a \\\sin b & 2.\cos^2b & \sin^2 b \\\sin c & 2.\cos^2c & \sin^2 c \\\end{vmatrix} \\=2\cdot \begin{vmatrix}\sin a & \cos^2a & \sin^2 a \\\sin b & \cos^2b & \sin^2 b \\\sin c & \cos^2c & \sin^2 c \\\end{vmatrix}\overset{K_2\to K_2+K_3}{=}2\cdot \begin{vmatrix}\sin a & 1 & \sin^2 a \\\sin b & 1 & \sin^2 b \\\sin c & 1 & \sin^2 c \\\end{vmatrix}\\=-2\cdot \begin{vmatrix}1 & \sin a & \sin^2 a \\1 & \sin b & \sin^2 b \\1 & \sin c & \sin^2 c \\\end{vmatrix}\overset{Vandermonde}{=} -2(\sin a - \sin b)(\sin b - \sin c)(\sin c - \sin a)$