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Algebra Level pending

Given: $$sin(x)\, +\, cos(x)\, =\, \frac { 1 }{ 2 }$$ And $${ sin }^{ 3 }(x)\, +\, { cos }^{ 3 }(x)\, =\, \frac { a }{ b }$$

Find: $$b\, -\, a$$ such that $$a$$ and $$b$$ are co-prime integers.

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