Constant Shenanigans!

Geometry Level 4

k=11000(k1+5)ϕ([(1+sin(kτ))(1sin(kτ)]π[12(eikτ+eikτ)]2e)= ?\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:

  • τ\tau is the almighty circle constant.

  • π\pi is the lesser circle constant.

  • ϕ\phi is the golden ratio.

  • ee is the base of the natural logarithm.

  • ii is the imaginary unit.

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

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