Constant Shenanigans!

Geometry Level 4

\[\large \sum_{k=1}^{1000} \sqrt{\sqrt[\phi]{(k^{1 + \sqrt{5}})}\left(\dfrac{\left[(1+\sin(k\tau))(1-\sin(k\tau)\right]^{\left \lfloor \pi \right \rfloor}}{\left[\frac{1}{2}(e^{ik\tau} + e^{-ik\tau})\right]^{2\left \lfloor e \right \rfloor}}\right)} = \ ? \]

Details and Assumptions:

  • \(\tau\) is the almighty circle constant.

  • \(\pi\) is the lesser circle constant.

  • \(\phi\) is the golden ratio.

  • \(e\) is the base of the natural logarithm.

  • \(i\) is the imaginary unit.

  • \(\lfloor x \rfloor\) is the greatest integer function.

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