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Find the exact value for $\sin { 112.5 }^{ \circ }$ by using the half angle formula.

A

$\cfrac{\sqrt{2-\sqrt{2}}}{4}$

B

$\cfrac{\sqrt{2-\sqrt{2}}}{2}$

C

$-\cfrac{\sqrt{2+\sqrt{2}}}{2}$

D

$\cfrac{\sqrt{2+\sqrt{3}}}{2}$

E

$\cfrac{\sqrt{2+\sqrt{2}}}{2}$

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