Animated Solution for Mathematics - Trigonometry: The value of $\cos \left(\frac{2\pi }{7}\right)+\cos \left(\frac{4\pi }{7}\right)+\cos \left(\frac{6\pi }{7}\right)$ is equal to :

The expression provided is $S=\cos (\frac{2\pi }{7})+\cos (\frac{4\pi }{7})+\cos (\frac{6\pi }{7})$. At first glance, this may appear intimidating, but the angles $\frac{2\pi }{7}$, $\frac{4\pi }{7}$, and $\frac{6\pi }{7}$ are not random.

They form an Arithmetic Progression with a common difference of $\theta =\frac{2\pi }{7}$. Recognizing this pattern is the 'Spark' of our solution.

Whenever you encounter a sum of cosines where the angles are in an A.P., you should utilize the standard summation formula:

Alternatively, using the general form for $n$ terms with first angle $\alpha$ and common difference $\theta$:

Here, we have $n=3$ terms, the first angle $\alpha =\frac{2\pi }{7}$, and the common difference $\theta =\frac{2\pi }{7}$. Substituting these values into the formula:

Simplifying the arguments, we obtain:

To resolve the product of sine and cosine, we multiply and divide by $2$ to invoke the product-to-sum identity $2\sin A\cos B=\sin (A+B)+\sin (A-B)$. The numerator becomes:

This simplifies to $\sin (\pi )+\sin (-\frac{\pi }{7})$. Since $\sin (\pi )=0$ and $\sin (-\theta )=-\sin (\theta )$, the expression reduces to:

The sine terms cancel out, leaving us with the elegant final result: