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calculus

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Verify the identity:

arctanx + arctan(1/x)=∏/2

using calculus theory.

(Hint: Differentiate the left hand side of the identity)

  • calculus -

    Actually, since tan(pi/2-x) = cot(x)
    and cot = 1/tan it falls right out.

    Or, using the sum of tangents formula,

    tan(arctanx + arctan(1/x))
    = [tan(arctan(x)) + tan(arctan(1/x))][1 - tan(arctan(x))*tan(arctan(1/x))]

    = [x + 1/x]/[1 - x*1/x] = (x + 1/x)/0 = oo

    tan pi/2 = oo

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