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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