Answer:
tan(22.5°) = √2 -1
Step-by-step explanation:
Using the half-angle formula for tangent, we can find the tangent of 45°/2 = 22.5°.
[tex]\tan{\dfrac{\theta}{2}}=\sqrt{\dfrac{1-\cos{\theta}}{1+\cos{\theta}}}\\\\\tan{\dfrac{45^\circ}{2}}=\sqrt{\dfrac{1-\cos{45^\circ}}{1+\cos{45^\circ}}}=\sqrt{\dfrac{1-\dfrac{1}{\sqrt{2}}}{1+\dfrac{1}{\sqrt{2}}}}=\sqrt{\dfrac{\sqrt{2}-1}{\sqrt{2}+1}}\\\\=\sqrt{\dfrac{\sqrt{2}-1}{\sqrt{2}+1}\cdot\dfrac{\sqrt{2}-1}{\sqrt{2}-1}}=\sqrt{\dfrac{(\sqrt{2}-1)^2}{2-1}}=\boxed{\sqrt{2}-1}[/tex]
