Geometry
# Trigonometric Functions

An ant finds itself trapped in the \(xy\)-plane, and its initial position is \((1,0).\)

Let \(S_k\) denote the circle with radius \(k\) centered around the origin. Starting from \((1,0)\), the ant walks 1 unit counter-clockwise on \(S_{1}.\) Then, it walks directly (radially outward) to \(S_2,\) on which it will walk 2 units counter-clockwise. Then, it will walk directly to \(S_3\) and walk 3 units counter-clockwise, and so, with the ant walking \(k\) units on \(S_k.\) (See the image above.)

When the ant crosses the positive \(x\)-axis for the first time since it left \((1,0)\), it is on \(S_{n}\). What is \(n\)?

Find the value of

\[\frac{\cos^4 75^{\circ}+\sin^4 75^{\circ}+3\sin^2 75^{\circ}\cos^2 75^{\circ}}{\cos^6 75^{\circ}+\sin^6 75^{\circ}+4\sin^2 75^{\circ}\cos^2 75^{\circ}}.\]

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