The sum of all the solutions of the equation cos⁡θcos⁡π3+θcos⁡π3−θ=14,θ∈[0,6π]

2cos⁡θcos⁡120∘+cos⁡2θ=1 or  2cos⁡θ−12+2cos2⁡θ−1=1 or  4cos3⁡θ−3cos⁡θ−1=0 or  cos⁡3θ=1=cos⁡0 or  3θ=2nπ or θ=2nπ3,n∈ZGiven the values so that 2n does not exceed 18. n=0, 1,2,3, ...,9Hence the sum  =2π3∑19 n=2π3×9(9+1)2=30π