How to prove whether the equation set has a unique solution? \begin{eqnarray}
\begin{cases}
\sin A \sin C-(\sin B)^2=0
\cr AC-B^2=0
\cr A+B+C-\pi=0
\cr A>0,B>0,C>0
\end{cases} \end{eqnarray} 
How to prove whether the equation set has a unique solution or not ?

The third and fourth conditions tell me you are investigating a triangle. So fixing points $A$ and $B$ of a triangle $ABC$, I plotted the first two conditions depending on where $C$ might lie:

The color code is as follows:
\begin{alignat*}{2}
\text{Blue:}&\qquad& \sin A\sin C-\sin^2B &> 0 &\qquad AC-B^2 &> 0 \\
\text{Magenta:}&& \sin A\sin C-\sin^2B &> 0 &\qquad AC-B^2 &< 0 \\
\text{Yellow:}&& \sin A\sin C-\sin^2B &< 0 &\qquad AC-B^2 &< 0
\end{alignat*}
So the inner boundary line between blue and magenta is where the second equation is satisfied, while the outer boundary line between magenta and yellow is where the first equation is satisfied. Both equations can only be satisfied simultaneously if the two curves touch, which they only do in the labeled point $C$, the corner of a regular triangle.
Now that I think about it, that plot is needlessly breaking the symmetry of the situation. I should fix $A$ and $C$ and plot depending on the position of $B$:

Of course, such a plot is not a proof, but it might help others in finding a proof. And for me personally it is enough to convince me that the regular triangle is indeed the only solution here.
For those interested, the plot displayed above was created with Cinderella, using its colorplot function.
