Show that the the triangle is isosceles

In a triangle ABC,if$ Cos A=\frac{sinB}{2sinC} $, then show that the the triangle is isosceles.

$$\frac{b^2+c^2-a^2}{2bc}=\frac{2R\sin B}{4R\sin C}=\frac{b}{2c}$$
$$c^2=a^2$$
$$c=a$$

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