The discussion focuses on proving the trigonometric identity involving the sum of cosines at specific angles. Participants suggest using angle addition formulas to express each cosine term in terms of known angles, specifically $\frac{\pi}{5}$ and $\frac{\pi}{7}$. There is a correction regarding the cosine of $\frac{18\pi}{35}$, emphasizing the importance of handling negative signs accurately. The next steps involve expanding the expressions, adding them, and factoring to equate coefficients. The conversation highlights the collaborative effort to clarify the proof process for the given identity.