arctan(x)-arccot(x)=(pi/4) solve for x without a calculator

Answer 1

We take tan of both parts:

tan(arctan(x) - arccot(x)) = tan (pi/4)

(tan(arctan(x) - tan(arccotan(x))/(1 +tan(arctan(x))*tan(arccot(x)) = 1;

(x - 1/x)/(1+x*1/x) =1, x - 1/x = 2, x^2 - 2x -1 = 0

x = 1 + sqrt(2)

or x = 1 - sqrt(2)

Answer 2

Arccotx = π/2 - Arctanx

So, if Arctanx - Arccotx = π/4, then Arctanx - (π/2 - Arctanx) = π/4

2Arctanx - π/2 = π/4

Arctanx = 3π/8

x = Tan(3π/8) = Tan [ (3π/4) / 2 ] = sin(3π/4) / [1 + cos(3π/4)] = (√2/2) / [1 - (√2/2)]

= √2 / (2 - √2)