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The value of $\left|\begin{array}{lll}\sin ^2 14^{\circ} & \sin ^2 66^{\circ} & \tan 135^{\circ} \\ \sin ^2 66^{\circ} & \tan 135^{\circ} & \sin ^2 14^{\circ} \\ \tan 135^{\circ} & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right|$ is
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$0$
Here,
$\left|\begin{array}{ccc}\sin ^2 14^{\circ} & \sin ^2 66^{\circ} & \tan 135^{\circ} \\ \sin ^2 66^{\circ} & \tan 135^{\circ} & \sin ^2 14^{\circ} \\ \tan 135^{\circ} & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right|$
Applying $C_1 \rightarrow C_1+C_2+C_3$
$=\left|\begin{array}{lll}\sin ^2 14^{\circ}+\sin ^2 66^{\circ}+\tan 135^{\circ} & \sin ^2 66^{\circ} \tan 135^{\circ} \\ \sin ^2 14^{\circ}+\sin ^2 66^{\circ}+\tan 135^{\circ} & \tan 135^{\circ} \sin ^2 14^{\circ} \\ \sin ^2 14^{\circ}+\sin ^2 66^{\circ}+\tan 135^{\circ} & \sin ^2 14^{\circ} \sin ^2 66^{\circ}\end{array}\right|$
$\begin{aligned} & =\sin ^2 14^{\circ}+\sin ^2 66^{\circ}+\tan 135^{\circ} \\ & \qquad\left|\begin{array}{lll}1 & \sin ^2 66^{\circ} & \tan 135^{\circ} \\ 1 & \tan 135^{\circ} & \sin ^2 14^{\circ} \\ 1 & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right|\end{aligned}$
$=\sin ^2 14^{\circ}+\sin ^2 66^{\circ}-1\left|\begin{array}{ccc}1 & \sin ^2 66 & -1 \\ 1 & -1 & \sin ^2 14^{\circ} \\ 1 & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right|$
Applying $R_1 \rightarrow R_1-R_2, R_2 \rightarrow R_2-R_3$
$\begin{aligned} & =\cos ^2 66^{\circ}+\sin ^2 66^{\circ}-1 \\ & \left|\begin{array}{ccc}0 & 1+\sin ^2 66^{\circ} & -1-\sin ^2 14^{\circ} \\ 0 & -1-\sin ^2 14^{\circ} & \sin ^2 14^{\circ}-\sin ^2 66^{\circ} \\ 1 & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right| \\ & \left(\cos ^2 66^{\circ}+\sin ^2 66^{\circ}-1\right)=0\end{aligned}$
$\left|\begin{array}{ccc}\sin ^2 14^{\circ} & \sin ^2 66^{\circ} & \tan 135^{\circ} \\ \sin ^2 66^{\circ} & \tan 135^{\circ} & \sin ^2 14^{\circ} \\ \tan 135^{\circ} & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right|$
Applying $C_1 \rightarrow C_1+C_2+C_3$
$=\left|\begin{array}{lll}\sin ^2 14^{\circ}+\sin ^2 66^{\circ}+\tan 135^{\circ} & \sin ^2 66^{\circ} \tan 135^{\circ} \\ \sin ^2 14^{\circ}+\sin ^2 66^{\circ}+\tan 135^{\circ} & \tan 135^{\circ} \sin ^2 14^{\circ} \\ \sin ^2 14^{\circ}+\sin ^2 66^{\circ}+\tan 135^{\circ} & \sin ^2 14^{\circ} \sin ^2 66^{\circ}\end{array}\right|$
$\begin{aligned} & =\sin ^2 14^{\circ}+\sin ^2 66^{\circ}+\tan 135^{\circ} \\ & \qquad\left|\begin{array}{lll}1 & \sin ^2 66^{\circ} & \tan 135^{\circ} \\ 1 & \tan 135^{\circ} & \sin ^2 14^{\circ} \\ 1 & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right|\end{aligned}$
$=\sin ^2 14^{\circ}+\sin ^2 66^{\circ}-1\left|\begin{array}{ccc}1 & \sin ^2 66 & -1 \\ 1 & -1 & \sin ^2 14^{\circ} \\ 1 & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right|$
Applying $R_1 \rightarrow R_1-R_2, R_2 \rightarrow R_2-R_3$
$\begin{aligned} & =\cos ^2 66^{\circ}+\sin ^2 66^{\circ}-1 \\ & \left|\begin{array}{ccc}0 & 1+\sin ^2 66^{\circ} & -1-\sin ^2 14^{\circ} \\ 0 & -1-\sin ^2 14^{\circ} & \sin ^2 14^{\circ}-\sin ^2 66^{\circ} \\ 1 & \sin ^2 14^{\circ} & \sin ^2 66^{\circ}\end{array}\right| \\ & \left(\cos ^2 66^{\circ}+\sin ^2 66^{\circ}-1\right)=0\end{aligned}$
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