Calculate the exact value for csc 22.5 degrees

I know that csc = 1/sin but I'm not really sure how to go from csc(22.5) = 1/(sin 22.5)

sin^2(x) = 1/2(1 - cos(2x))

Letting x be 22.5 degrees gives:

sin^2(22.5) = 1/2(1 - cos(45))

sin^2(22.5) = 1/2(1 - √2//2)

sin(22.5) = √(1/2(1 - √2/2))

csc(22.5) = 1 / √(1/2(1 - √2/2))

sin(22.5) =>
sin(45/2) =>
sqrt((1/2) * (1 - cos(45))) =>
sqrt((1/2) * (1 - sqrt(2)/2)) =>
sqrt((1/2) * (1/2) * (2 - sqrt(2))) =>
(1/2) * sqrt(2 - sqrt(2))

csc(22.5) =>
1/sin(22.5) =>
1/((1/2) * sqrt(2 - sqrt(2))) =>
2 / sqrt(2 - sqrt(2)) =>
2 * sqrt(2 - sqrt(2)) / (2 - sqrt(2)) =>
2 * (2 + sqrt(2)) * sqrt(2 - sqrt(2)) / (4 - 2) =>
2 * (2 + sqrt(2)) * sqrt(2 - sqrt(2)) / 2 =>
(2 + sqrt(2)) * sqrt(2 - sqrt(2))

sin 2x= 2 sin x cos x
x=22.5
cos2x = 1 − 2 sin2 x
cos 45 = 1 − 2 sin2 22.5
1/ sqrt2 = 1 − 2 sin2 22.5
2 sin2 22.5=1- 1/ sqrt2 = 0.2929
sin2 22.5= 0.2929/2=0.14645
sin 22.5= +/- 0.38268
csc 22.5= 1/sin 22.5 =+/-2.6131