If the value of $\cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{6\pi}{7} + \cos\frac{7\pi}{7} = -\frac{l}{2}$. Find the value of $l$
Step-by-Step Solution
Key Concept: Sum-of-cosines formulas for roots of unity simplify complex trigonometric expressions.
Using the identity $\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} = -\frac{1}{2}$ and simplifying the given expression with $\sin \frac{3\pi}{7}$ in numerator and denominator yields $\cos \frac{4\pi}{7} - 1 = -\frac{3}{2}$.
Correct Answer: 3