Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

From conditional identities, $\frac{\sin 2A+\sin 2B+\sin 2C}{\sin A+\sin B+\sin C}=\frac{4\sin A\sin B\sin C}{4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}}=\frac{8\sin \frac{A}{2}\cos \frac{A}{2}\sin \frac{B}{2}\cos \frac{B}{2}\sin \frac{C}{2}\cos \frac{C}{2}}{\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}}=8\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$. Hence $k=8$.