Why is it stairway to heaven? Can't there be an elevator?

Calculus Level 4

A stair case has the property such that it travels in the pattern right, up, right, up, right, up. . . ad infinitum.

It travels \(\cos^1\left(\frac{\pi}{10}\right)\) to the right, \(\cos^2\left(\frac{\pi}{10}\right)\) up, \(\cos^3\left(\frac{\pi}{10}\right)\) right, \(\cos^4\left(\frac{\pi}{10}\right)\) up. . .

Find the total length of displacement from the beginning of this staircase to the end.

If the answer can be represented in the form \(\sqrt{a+b\sqrt{c}}\). Find \(a+b+c\)

Inspired by my good friend Brian Charlesworth

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