Find the exact value of sin22.5 (using the half angle formula) i have the answer from the book, but i don't understand how they got there

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Find the exact value of sin22.5 (using the half angle formula) i have the answer from the book, but i don't understand how they got there

Mathematics
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half angle formula is obtained by manipulating the cos(2a) formula
cos(2a) has 3 equal solutions: cos^2 - sin^2 1 - 2sin^2 2cos^2 -1
cos(2a) = 1-2sin^2 is helpful in this case.. cos(2a) -1 = 2sin^2 cos(2a) -1 -------- = sin^2 2

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yes, i just don't understand how they simplified this: sqrt ((1 - [sqrt2]/2) / 2) it is supposed to become sqrt ((2 - sqrt2) / 4 )
they had steps, and it didn't explain how they got from step to step
ohhhh......
so you wanna see it simplified then right?
it just said "common denominator" and i couldn't figure out how to simplify it
i guess yes, i just want it explained really
sqrt(1 - cos(45) /2) = sin(22.5) right? sqrt([1 - sqrt(2)/2]/2) we can seperate the "square root" sign to cover the top and bottom of the fraction seperately... for example: sqrt(a/b) = sqrt(a) / sqrt(b) right?
let me write it out this way so I dont confuse meself: sqrt(2) 1 - ----- 2 ----------- 2
2 sqrt(2) -- - ------- 2 2 ------------- 2
2 - sqrt(2) --------- 2 ------------ 2
2-sqrt(2) 1 -------- x --- 2 2
2 - sqrt(2) --------- 4 Do you follow that?
hold on
i'm confused. i thought you multiplied 2-sqrt(2) 2 -------- x --- 2 1 because 2 is the reciprocal isnt it?
or am i completely wrong
lets take a look at a simpler example: 1/2 --- = 1/4 right? 2
whenever we divide by a fraction we turn it over an multiply instead. 1/2 --- = 1/2 * 1/2 = 1/4 2 1/3 --- = 1/3 * 1/2 = 1/6 2
7 --- = 7 * (3/1) = 21 1/3
ohhhhhh got it thnks!
I knew you would :)
thank you!

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