(xyz) sin 3$\mathrm{\theta }$ – y cos 3$\mathrm{\theta }$ – z cos 3$\mathrm{\theta }$ = 0                               ....(i)

(xyz) sin 3$\mathrm{\theta }$ – (2 sin 3$\mathrm{\theta }$) y – (2cos 3$\mathrm{\theta }$)  z = 0                    ....(ii)

(xyz) sin 3$\mathrm{\theta }$ – (cos 3$\mathrm{\theta }$ + sin 3$\mathrm{\theta }$)y  –(2 cos 3$\mathrm{\theta }$)z = 0            ....(iii)

 $\left|\begin{array}{ccc}\sin 3\mathrm{\theta }& \cos 3\mathrm{\theta }& \cos 3\mathrm{\theta }\\ \sin 3\mathrm{\theta }& 2\sin 3\mathrm{\theta }& 2\cos 3\mathrm{\theta }\\ \sin 3\mathrm{\theta }& \cos 3\mathrm{\theta }+\sin 3\mathrm{\theta }& 2\cos 3\mathrm{\theta }\end{array}\right|=0$

 $\Rightarrow \sin 3\mathrm{\theta }.\cos 3\mathrm{\theta }\left|\begin{array}{ccc}1& \cos 3\mathrm{\theta }& 1\\ 1& 2\sin 3\mathrm{\theta }& 2\\ 1& \cos 3\mathrm{\theta }+\sin 3\mathrm{\theta }& 2\end{array}\right|=0$

$\Rightarrow \sin 3\mathrm{\theta }.\cos 3\mathrm{\theta }\left|\begin{array}{ccc}1& \cos 3\mathrm{\theta }& -1\\ 1& 2\sin 3\mathrm{\theta }& 0\\ 1& \cos 3\mathrm{\theta }+\sin 3\mathrm{\theta }& 0\end{array}\right|=0$

$\Rightarrow$ sin 3$\mathrm{\theta }$ . cos 3$\mathrm{\theta }$ (cos 3$\mathrm{\theta }$ + sin 3$\mathrm{\theta }$ – 2 sin3$\mathrm{\theta }$) = 0

 $\Rightarrow$ sin 3$\mathrm{\theta }$ . cos 3$\mathrm{\theta }$ (cos 3$\mathrm{\theta }$ – sin 3$\mathrm{\theta }$ )= 0 ......Equation (A)

From given equations,

if sin 3$\mathrm{\theta }$ = 0 then equation (2) becomes

 $\frac{2\cos 3\mathrm{\theta }}{\mathrm{y}}=0\Rightarrow \mathrm{No}\mathrm{solution}$

  $\therefore \sin 3\mathrm{\theta }\ne 0$

Similarly, if cos 3$\mathrm{\theta }$ = 0 then equation (1) $\Rightarrow$ x = 0

for which equation (2) $\Rightarrow 0=\frac{2\sin 3\mathrm{\theta }}{\mathrm{z}}$  which is not possible

 $\therefore$  from equation (A)

 cos 3$\mathrm{\theta }$ – sin 3$\mathrm{\theta }$ = 0

 or tan 3$\mathrm{\theta }$ = 1,

 $\Rightarrow 3\mathrm{\theta }=\frac{\mathrm{\pi }}{4},\frac{5\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{4}$

 $\therefore \mathrm{\theta }=\frac{\mathrm{\pi }}{12},\frac{5\mathrm{\pi }}{12},\frac{7\mathrm{\pi }}{12}$

 $\therefore$ Number of possible values of $\mathrm{\theta }$ in (0, $\mathrm{\pi }$) = 3