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For x in (0,1), let f(x) = sec( arctan[ (sin(arccos x) + cos(arcsin x)) / (cos(arccos x) + sin(arcsin x)) ]). Evaluate the sum from r = 2 to r = 10 of f(1/r).

1. 54
2. 45
3. 55
4. 44

Correct answer: 54

Solution

Both sin(arccos x) and cos(arcsin x) equal sqrt(1 - x²), so the numerator is 2*sqrt(1 - x²). Both cos(arccos x) and sin(arcsin x) equal x, so the denominator is 2x. The ratio is sqrt(1 - x²)/x. Then sec(arctan(t)) = sqrt(1 + t²) = sqrt(1 + (1 - x²)/x²) = 1/x. So f(x) = 1/x, hence f(1/r) = r. The required sum is 2 + 3 +... + 10 = 54.

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