Answer
$sin(22.5^o)=\frac{\sqrt{2-\sqrt{2}}}{2}$
Work Step by Step
Remember$\;\;\;\;\;sin(\frac{x}{2})=\pm \sqrt{\frac{1-cos(x)}{2}}$
$sin(22.5^o)=\pm \sqrt{\frac{1-cos(45^o)}{2}}$
We know $22.5^o$ in quadrant $I$ so $sin(22.5^o)$ is positive
$sin(22.5^o)= \sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}$
$sin(22.5^o)= \sqrt{\frac{2-\sqrt{2}}{4}}$
$sin(22.5^o)=\frac{\sqrt{2-\sqrt{2}}}{2}$
