1. Stated problem: Simplify the expression $$2r\cdot\cos\left(\frac{a}{2}\right)\cdot \frac{\sin(a)}{\sin\left(\frac{a}{2}\right)\sin\left(\frac{3a}{2}\right)}.$$\n\n2. Use the double-angle identity for sine: $$\sin(a) = 2\sin\left(\frac{a}{2}\right)\cos\left(\frac{a}{2}\right).$$\n\n3. Substitute into the expression: $$2r\cdot \cos\left(\frac{a}{2}\right) \cdot \frac{2\sin\left(\frac{a}{2}\right)\cos\left(\frac{a}{2}\right)}{\sin\left(\frac{a}{2}\right)\sin\left(\frac{3a}{2}\right)}.$$\n\n4. Simplify numerator: $$2r \cdot \cos\left(\frac{a}{2}\right) \cdot 2\sin\left(\frac{a}{2}\right)\cos\left(\frac{a}{2}\right) = 4r \sin\left(\frac{a}{2}\right) \cos^{2}\left(\frac{a}{2}\right).$$\n\n5. Cancel $$\sin\left(\frac{a}{2}\right)$$ in numerator and denominator: expression becomes $$\frac{4r \cos^{2}\left(\frac{a}{2}\right)}{\sin\left(\frac{3a}{2}\right)}.$$\n\n6. Final simplified expression: $$\boxed{\frac{4r \cos^{2}\left(\frac{a}{2}\right)}{\sin\left(\frac{3a}{2}\right)}}.$$