Answer:
√2 - 1
Explanation:
you can use the half angle formula to solve this question.
because we aren't using a calculator, we want to find a degree that has easy values, like 45, 30, or 60 degrees. 22.5 *2 = 45 degrees, so we can use the half angle formula.
tan(A/2) = sin(A)/(1+cosA)
while there are multiple forms of the tangent half angle formula, I'm just sticking with this one.
tan(45/2) = sin(45)/(1+cos45)
= (√2/2)/(1+√2/2)
= (√2/2)/( (2+√2)/2 )
= (√2/2) * (2/(2+√2))
= √2/(2+√2)
Now to simplify completely, we should rationalize the denominator (remove the radicals from the denominator) by multiplying by (2-√2) on both the denominator and numerator.
(√2/(2+√2)) * (2-√2)/(2-√2)
= (√2 (2-√2)) / (4 - 2) = (2√2 - 2)/2 = √2 - 1
Please give a thanks if that helped :)