Let θ ∈ (0, (π)/(2)) . If the system of linear equations (1 + ^ 2...

\left|$\begin{cases}1 + \cos^{2}\theta & \sin^{2}\theta & 4\sin 3\theta \\ \cos^{2}\theta & 1 + \sin^{2}\theta & 4\sin 3\theta \\ \cos^{2}\theta & \sin^{2}\theta & 1 + 4\sin 3\theta\end{cases}$\right| = 0$</p><p>${R}_{3} \to {R}_{3} - {R}_{2}</p><p>\left|\begin{cases}1 + \cos^{2}\theta & \sin^{2}\theta & 4\sin 3\theta \\ \cos^{2}\theta & 1 + \sin^{2}\theta & 4\sin 3\theta \\ 0 & - 1 & 1\end{cases}\right|

${C}_{2} \to {C}_{2} + {C}_{1}$

\left|$\begin{cases}1 + \cos^{2}\theta & 2 & 4\sin 3\theta \\ \cos^{2}\theta & 2 & 4\sin 3\theta \\ 0 & - 1 & 1\end{cases}$\right|$</p><p>${C}_{2} \to {C}_{2} + {C}_{3}</p><p>\left|\begin{cases}1 + \cos^{2}\theta & 2 + 4\sin 3\theta & 4\sin 3\theta \\ \cos^{2}\theta & 2 + 4\sin 3\theta & 4\sin 3\theta \\ 0 & 0 & 1\end{cases}\right| = 0

Expanding along ${R}_{3}$

$⇒ \left(1 + \cos^{2}\theta\right)\left(2 + 4\sin 3\theta\right) - \left(2 + 4\sin 3\theta\right)\left(\cos^{2}\theta\right) = 0$

$⇒ \left(2 + 4\sin 3\theta\right)\left(1 + \cos^{2}\theta - \cos^{2}\theta\right) = 0$

$⇒ \sin 3\theta = \dfrac{- 1}{2}$

$⇒ 3\theta = \dfrac{7\pi}{6}$

$⇒ \theta = \dfrac{7\pi}{18}$