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Consider the equation 2*arcsin(x) + 2*arccos(x) + 2*arctan(x) = 3*pi. The number of real solutions of this equation is less than which of the following values?

1. 0
2. 1
3. 2
4. 3

Correct answer: 2

Solution

The identity arcsin(x) + arccos(x) = pi/2 holds for x in [-1, 1]. Substituting reduces the equation to pi + 2*arctan(x) = 3*pi, so arctan(x) = pi, which is impossible since arctan(x) in (-pi/2, pi/2). Hence there are 0 real solutions, and 0 < 2.

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