Question

cosx/11+cos3x/11+cos5x/11+cos7x/11+cos9x/11

Answer 1

cosA+cosB=2cos(A+B/2)cos(A-B/2) And cos(-x)=cosx are required formulas for this.
cos x/11+cos9x/11= 2cos(5x/11)cos(-4x/11)..(1)
cos 3x/11+cos 7x/11=2cos(5x/11)cos(-2x/11)..(2). cos(5x/11)..(3)
So add three quantities (1),(2) and(3).
take cos(5x/11) common from three then we will get
cos(5x/11)(2cos(4x/11)+2cos(2x/11)+1)
=> cos(5x/11)(4 cos (3x/11)cos(x/11) +1).