# Can you deduce this?

Calculus Level 5

$\large \int \sqrt{\dfrac{1-\cos x}{1 + \cos x}} \left ( \prod_{r=0}^n (1 + \sec (2^r x) ) \right) \, dx$

Let $$f_{(n)}$$ be a function as described above for non-negative integer $$n$$. Let $$g(x) = f_{(4)} (x)$$. If the solution to $$g(x) =1$$ can be written in the form of $$\dfrac1{16} \arccos (e^k)$$, find $$(|k| - 9)!$$.

$$f_{(n)}(0)=0$$, and take $$\cos x \ne -1, \cos (nx) \ne 0$$.

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