Solve the equation tan^-1 (x-1/x-2 ) tan^-1 (2x-1/2x 1) = tan^-1 23/36

Answer:

x = -3/8  & x = 4/3

Step-by-step explanation:

Tan⁻¹(x-1/x+1)+Tan⁻¹(2x-1/2x+1)=Tan⁻¹(23/36)

Let say Tan⁻¹((x-1)/(x+1)) = a  => Tan a = (x-1)/(x+1)

Tan⁻¹((2x-1)/(2x+1)) = b  => Tan b = (2x-1)/(2x+1)

=> a + b = Tan⁻¹(23/36)  - eq 1

Using formula

Tan ( a + b) = (Tan a + Tan b )/( 1 - Tan a . Tan b)

=> Tan  ( a + b) = ((x-1)/(x+1) + (2x-1)/(2x+1)) / ( 1  - ((x-1)/(x+1))((2x-1)/(2x+1) )

=> Tan  ( a + b) =( (x-1)(2x+1) + (2x-1)(x+1) ) / ( (2x+1)(x+1) - (2x-1)(x-1) )

=> Tan  ( a + b) = (2x² -x -1 + 2x² + x  - 1) / ( 2x² + 3x + 1 - (2x² -3x + 1) )

=> Tan  ( a + b) = (4x² - 2)/6x

=> Tan  ( a + b) = (2x² - 1)/3x

=> a + b = Tan⁻¹((2x² - 1)/3x)

Equating with eq 1

=> Tan⁻¹((2x² - 1)/3x) = Tan⁻¹(23/26)

=> (2x² - 1)/3x = 23/36

=>  (2x² - 1) = 23x/12

=> 24x² - 12 = 23x

=> 24x² - 23x - 12 = 0

=> 24x² -32x + 9x - 12 = 0

=> 8x(3x - 4) + 3(3x -4) = 0

=> (8x +3)(3x -4) = 0

=> x = -3/8  & x = 4/3