The number of solutions of ∑r=15 cos⁡rx=5 in the interval [0,2π] is

∑r=15 cos⁡rx=5⇒ cos⁡x+cos⁡2x+cos⁡3x+cos⁡4x+cos⁡5x=5which is possible only when cosx= cos 2x= cos3x= cos 4x= cos5x = 1 and is satisfied by x = 0 and x = 2π