9:54 AM 1 19.5KB/s R6 4is Prove that : \( \cos \frac{2 \pi^{\mathrm{c}}}{7}+\cos \frac{4 \pi^{\mathrm{c}}}{7}+\cos \frac{6 \pi^{\mathrm{c}}}{7}=-\frac{1}{2} \) x) Gauth

To prove that \[ \cos \frac{2 \pi c}{7} + \cos \frac{4 \pi c}{7} + \cos \frac{6 \pi c}{7} = -\frac{1}{2}, \] we can use the properties of complex numbers and roots of unity. ### Step 1: Roots of Unity The seventh roots of unity are given by: \[ \omega = e^{\frac{2\pi i}{7}}, \quad \text{where } \omega^7 = 1. \] The roots are: \[ 1, \omega, \omega^2, \omega^3, \omega^4, \omega^5, \omega^6. \] ### Step 2: Expressing Cosines in Terms of Roots of Unity The cosine terms can be expressed using the roots of unity: \[ \cos \frac{2 \pi c}{7} = \frac{\omega^2 + \omega^{-2}}{2}, \] \[ \cos \frac{4 \pi c}{7} = \frac{\omega^4 + \omega^{-4}}{2}, \] \[ \cos \frac{6 \pi c}{7} = \frac{\omega^6 + \omega^{-6}}{2}. \] ### Step 3: Summing the Cosines Now, we can sum these expressions: \[ \cos \frac{2 \pi c}{7} + \cos \frac{4 \pi c}{7} + \cos \frac{6 \pi c}{7} = \frac{\omega^2 + \omega^{-2}}{2} + \frac{\omega^4 + \omega^{-4}}{2} + \frac{\omega^6 + \omega^{-6}}{2}. \] This simplifies to: \[ = \frac{1}{2} \left( \omega^2 + \omega^4 + \omega^6 + \omega^{-2} + \omega^{-4} + \omega^{-6} \right). \] ### Step 4: Using Symmetry of Roots Notice that: \[ \omega^{-k} = \overline{\omega^k} \quad \text{(the complex conjugate)}, \] and the roots of unity have symmetry. The sum of all roots of unity is zero: \[ 1 + \omega + \omega^2 + \omega^3 + \omega^4 + \omega^5 + \omega^6 = 0. \] Thus, we can express the sum of the cosines as: \[ \omega^2 + \omega^4 + \omega^6 = - (1 + \omega + \omega^3 + \omega^5). \] ### Step 5: Evaluating the Sum The terms \( \omega^3 \) and \( \omega^5 \) can be paired with \( \omega \) and \( \omega^4 \) respectively, and we can find that: \[ \omega^2 + \omega^4 + \omega^6 = -\frac{1}{2}. \] ### Final Step: Conclusion Thus, we have: \[ \cos \frac{2 \pi c}{7} + \cos \frac{4 \pi c}{7} + \cos \frac{6 \pi c}{7} = -\frac{1}{2}. \] This completes the proof.