<s>[INST] If $A$, $B$ and $C$ are the angles of a triangle then find the value of $\Delta$ I'll state the question from my book below:

If $A$, $B$ and $C$ are the angles of a triangle, then find the determinant value of
  $$\Delta = \begin{vmatrix}\sin^2A & \cot A & 1 \\ \sin^2B & \cot B & 1 \\ \sin^2C & \cot C & 1\end{vmatrix}.$$

Here's how I tried solving the problem:
$\Delta = \begin{vmatrix}\sin^2A & \cot A & 1 \\ \sin^2B & \cot B & 1 \\ \sin^2C & \cot C & 1\end{vmatrix}$
$R_2 \to R_2 - R_1$
$R_3 \to R_3 -R_1$
$= \begin{vmatrix}\sin^2A & \cot A & 1 \\ \sin^2B-\sin^2A & \cot B-\cot A & 0 \\ \sin^2C-\sin^2A & \cot C-\cot A & 0\end{vmatrix}$
Expanding the determinant along $C_3$
\begin{align}
&= (\sin^2B-\sin^2A)(\cot C-\cot A)-(\cot B-\cot A)(\sin^2C-\sin^2A)  \\
&= \sin(B+A) \sin(B-A) \left[\frac {\cos C}{\sin C} - \frac {\cos A}{\sin A}\right] - \left[\frac {\cos B}{\sin B} - \frac {\cos A}{\sin A}\right]\sin(C+A) \sin(C-A)  \\
&= \frac {\sin(B+A) \sin(B-A) \sin(A-C)} {\cos A \cos C} - \frac {\sin(A-B) \sin(C+A) \sin(C-A)} {\cos A \cos C}  \\
&= \frac {\sin(B-A) \sin (A-C)} {\cos A} \left[\frac {\sin(A+B)} {\cos C} - \frac {\sin(A+C)} {\cos B}\right]  \\
&= \frac {\sin(B-A) \sin (A-C)} {\cos A} \left[\frac {\sin C} {\cos C} - \frac {\sin B} {\cos B}\right]  \\
&= \frac {\sin(B-A) \sin (A-C) \sin (C-B)} {\cos A \cos B \cos C}
\end{align}
I tried solving further but the expression just got complicated. I don't even know if the work I've done above is helpful. My textbook gives the answer as $0$. I don't have any clue about getting the answer. Any help would be appreciated.
 [/INST] $$F=\begin{vmatrix}  \sin^2B-\sin^2A & \cot B-\cot A   \\ \sin^2C-\sin^2A & \cot C-\cot A  \end{vmatrix}$$
$$=\begin{vmatrix}  \sin^2B-\sin^2A & -\dfrac{\sin(B-A)}{\sin A\sin B}   \\ \sin^2C-\sin^2A & -\dfrac{\sin(C-A)}{\sin C\sin A}  \end{vmatrix}$$
$$=\dfrac1{\sin B\sin^2A\sin C}\begin{vmatrix}\sin(B-A)\sin(B+A)\sin B\sin A&-\sin(B-A)\\ \sin(C-A)\sin(C+A)\sin C\sin A&-\sin(C-A)\end{vmatrix}$$
Using $A+B+C=\pi,\sin(B+A)=\sin C$ etc.,
$$F=\dfrac{\sin A\sin B\sin C}{\sin^2A\sin B\sin C}\begin{vmatrix}\sin(B-A)&-\sin(B-A)\\\sin(C-A)&-\sin(C-A)\end{vmatrix}=?$$
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