7.53 In $\triangle A B C$, if $2 \sin A \cos B=\sin C$, then this triangle is
(A) an isosceles triangle.
(B) a right triangle.
(C) a scalene triangle.
(D) an equilateral triangle.
(Anhui Province Wuhu City Junior High School Mathematics Competition, 1990)
[Solution] Given $A+B+C=180^{\circ}$, then $\sin C=\sin (A+B)$.
Also, from the given condition, we have $\sin (A+B)=2 \sin A \cos B$, which means
$$
\begin{array}{ll} 
& \sin A \cos B=\cos A \sin B, \text { or } \tan A=\tan B . \\
\because \quad 0^{\circ}<A, B<180^{\circ}, \therefore \quad A=B .
\end{array}
$$

Therefore, the answer is $(A)$.