A geometry problem by neelesh vij

Geometry Level 3

$\cos(x) \cos(2x) + \cos(2x) \cos(3x) + \cdots + \cos((n-1)x) \cos(nx) + \cos(nx) \cos(x) = n + \sin^2(nx)$

Find the number of solutions for the equation above, where $x \in [0,2 \pi]$ for a given value of $n$ where, $n \geq 2$ .

1 solution

Neelesh Vij
Feb 20, 2017

Notice that the LHS has a maximum value of $n$ as $\cos(A) \times \cos(B)$ has max value of $1$ and RHS has minimum value of $n$

So $x = 2k \pi$ where k is a whole number.

Please do not let your notation do double duty. Is the $n$ in your solution (both the first and second lines) equal to the $n$ in the equation?

Calvin Lin Staff - 4 years, 3 months ago

thanks for pointing out. i have done the necessary changes.

neelesh vij - 4 years, 3 months ago