Question

If $x+y+z=\pi$ and
$\Delta =\left|\begin{array}{ccc}\sin 3x& \sin 3y& \sin 3z\\ \sin x& \sin y& \sin z\\ \cos x& \cos y& \cos z\end{array}\right|$
then $\Delta$ equals

• A. $3$
• B. $-1$
• C. $0$
• D. $4$

Answer 1

Answer: (C)

$0$

$\Delta =\left|\begin{array}{ccc}\sin 3x& \sin 3y& \sin 3z\\ \sin x& \sin y& \sin z\\ \cos x& \cos y& \cos z\end{array}\right|$

$=\left|\begin{array}{ccc}3\sin x-4{\sin }^{3}x& 3\sin y-4{\sin }^{3}y& 3\sin z-4{\sin }^{3}z\\ \sin x& \sin y& \sin z\\ \cos x& \cos y& \cos z\end{array}\right|$

$=\left|\begin{array}{ccc}3\sin x& 3\sin y& 3\sin z\\ \sin x& \sin y& \sin z\\ \cos x& \cos y& \cos z\end{array}\right|-\left|\begin{array}{ccc}4{\sin }^{3}x& 4{\sin }^{3}y& 4{\sin }^{3}z\\ 0& 0& 0\\ 0& 0& 0\end{array}\right|$

$=3sinx(sinycosz-sinzcosy)-3siny(sinxcosz-sinzcosx)+3sinz(sinxcosy-sinycosx)-0$

$=0$