# A Tricky Trigonometry Question!!

Geometry Level 4

Let $$A$$ ,$$B$$ ,$$C$$ Be real Numbers Such That

(i) $$\left( \sin { A } ,\cos { B } \right)$$ Lies On a unit circle centred at Origin.

(ii) $$\tan { C }$$ and $$\cot { C }$$ are Defined.

If minimum value of $${ \left( \tan { C } -\sin { A } \right) }^{ 2 }+{ \left( \cot { C } -\cos { B } \right) }^{ 2 }$$ is $$\alpha +\sqrt { 2 } \beta$$.

Where $$\alpha , \beta \in \mathbb{I}$$

Then Find $${ \alpha }^{ 3 }+{ \beta }^{ 3 }$$.

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