tan^-1 (x-1/x+1)+tan^-1 (2x-1/2x+1)=tan^-1 (23/36)
Answers
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