Express cot 2θ in terms of cot θ

cot 2θ = 1/tan2θ
= (1-tan² θ)/2tanθ <----------------------公式
=(cot² θ-1)/2cotθ <----------------------分子分母同時除cot²θ

cot45 = (cot² 22.5 - 1) / 2 cot22.5 <-------------θ 代入22.5
1 = (cot² 22.5 - 1) / 2 cot22.5
2cot22.5 = (cot² 22.5 - 1)
0 = cot² 22.5 - 2 cot22.5 - 1
cot22.5 = [2±√(2² -4*1*(-1))]/2 <-------------求根公式
= 1±√2
= 1+ √2 <------------因大於零

cot² θ + 1 = csc² θ <------------公式
cot² 22.5 + 1 = csc² 22.5
( 1+ √2 )² +1 = csc² 22.5
3 + 2√2 + 1 = csc² 22.5
csc² 22.5 = 4 + 2√2

cot 2θ
= 1 / tan2θ
= (1 - tan^2 θ) / (2tanθ)
= (cot^2 θ - 1) / (2cotθ)

cot45 = (cot^2 θ - 1) / (2cotθ) where θ = 22.5∘
2cotθ = cot^2 θ - 1
cot^2 θ - 2cotθ - 1 = 0
cotθ = 1 + sqrt(2)

csc² 22.5∘
= 1 + cot² 22.5∘
= 1 + (1 + sqrt(2))^2
= 4 + 2sqrt(2)