We can use double angle identity cot2A=(cot^2A-1)/(2cotA)cot2A=cot2A−12cotA
As such cot8theta=(cot^2 4theta-1)/(2cot 4theta)cot8θ=cot24θ−12cot4θ
= (((cot^2 2theta-1)/(2cot2theta))^2-1)/(2(cot^2 2theta-1)/(2cot2theta))(cot22θ−12cot2θ)2−12cot22θ−12cot2θ
= ((cot^2 2theta-1)^2-(2cot2theta)^2)/(4cot2theta(cot^2 2theta-1)(cot22θ−1)2−(2cot2θ)24cot2θ(cot22θ−1)
= (cot^4 2theta+1-2cot^2 2theta-4cot^2 2theta)/(4cot^3 2theta-4cot2theta)cot42θ+1−2cot22θ−4cot22θ4cot32θ−4cot2θ
= (cot^4 2theta-6cot^2 2theta+1)/(4cot^3 2theta-4cot2theta)cot42θ−6cot22θ+14cot32θ−4cot2θ
= (((cot^2theta-1)/(2cottheta))^4-6((cot^2theta-1)/(2cottheta))^2+1)/(4((cot^2theta-1)/(2cottheta))^3-4((cot^2theta-1)/(2cottheta)))(cot2θ−12cotθ)4−6(cot2θ−12cotθ)2+14(cot2θ−12cotθ)3−4(cot2θ−12cotθ)
= ((cot^2theta-1)^4-24cot^2theta(cot^2theta-1)^2+(2cottheta)^4)/(8cottheta(cot^2theta-1)^3-32cot^3theta(cot^2theta-1))(cot2θ−1)4−24cot2θ(cot2θ−1)2+(2cotθ)48cotθ(cot2θ−1)3−32cot3θ(cot2θ−1)
= ((cot^8theta-4cot^6theta+6cot^4theta-4cot^2theta+1)-24cot^2theta(cot^4theta-2cot^2theta+1)+4cot^4theta)/(8cottheta(cot^6theta-3cot^4theta+3cot^2theta-1)-32cot^5theta+32cot^3theta)(cot8θ−4cot6θ+6cot4θ−4cot2θ+1)−24cot2θ(cot4θ−2cot2θ+1)+4cot4θ8cotθ(cot6θ−3cot4θ+3cot2θ−1)−32cot5θ+32cot3θ
= (cot^8theta-28cot^6theta+58cot^4theta-28cot^2theta+1)/(8cot^7theta-56cot^5theta+56cot^3theta-8cottheta)cot8θ−28cot6θ+58cot4θ−28cot2θ+18cot7θ−56cot5θ+56cot3θ−8cotθ
