prove that

cosθ+cos2θ+cos3θ=1/4

2cos2θcosθ+cos2θ=1/4

cos2θ(2cosθ+1)=1/4

(2cos²θ-1)(2cosθ+1)=1/4

4cos³θ+2cos²θ-2cosθ-5/4 = 0

let cosθ = t

16t³+8t²-8t-5 = 0

The only real root of this equation is 0.74.

hence, cosθ = 0.74

hence, θ = cos^-1(0.74)