tan inversex =-π+cot inverse 1/x=sin Inversex/root I+X2 =-Cosinverse I/root I+ x2 when X

There are two cases for this problem

Case 1 : x > 0

Case 2 : x

Case 1:

x > 0

let theta = tan^-1(1/x) where theta = (0, pi/2) i.e. not including 0 and pi/2

Then

tan(theta) = 1/x

cot(theta) = x

When x > 0,

theta = cot^-1(x)

Case 2:

x

Let theta = tan^-1(1/x) where theta = (pi/2, pi)

Then

tan(theta) = 1/x

-tan(pi - theta) = 1/x

tan(theta - pi) = 1/x

cot(theta - pi) = x

theta - pi = cot^-1(-x)

theta = -pi + cot^-1(-x)

Therefore,

tan^-1(1/x) = -pi + cot^-1(-x)