Solve: (i) arctan((x-1)/(x+1)) + arctan((2x-1)/(2x+1)) = arctan(23/36); (ii) 2 arctan(x) = arccos((-a²)/(1+a²)) - arccos((-b²)/(1+b²)), where a 0 and b 0.

Question

Solve: (i) arctan((x-1)/(x+1)) + arctan((2x-1)/(2x+1)) = arctan(23/36); (ii) 2 arctan(x) = arccos((-a²)/(1+a²)) - arccos((-b²)/(1+b²)), where a > 0 and b > 0.

Accepted Answer

(i) x = 2; (ii) x = (a - b)/(1 + ab). (i) Combine the two arctans with the addition formula and set equal to arctan(23/36); solving the resulting equation gives x = 2. (ii) Each arccos term reduces to a 2 arctan expression; the difference simplifies via the arctan subtraction formula to 2 arctan((a - b)/(1 + ab)), so x = (a - b)/(1 + ab).

Solve: (i) arctan((x-1)/(x+1)) + arctan((2x-1)/(2x+1)) = arctan(23/36); (ii) 2 arctan(x) = arccos((-a²)/(1+a²)) - arccos((-b²)/(1+b²)), where a > 0 and b > 0.

Solution

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