Question

Write the value of tan⁻¹ x + tan⁻¹ `(1/x)`  for x < 0.

Answer 1

`tan^-1x+tan^-1y=tan^-1((x+y)/(1-xy))`

When  `x<0,1/x<0,` then both are negative.

Let x = -y,  y > 0

Then,

`tan^-1x+tan^-1  1/x=tan^-1 (-y)+tan^-1(-1/y)`

`=-(tan^-1y+tan^-1  1/y)`

`=-tan^-1((y+1/y)/(1-y1/y)), y>0`

`=-tan^-1((y^2+1)/0)`

`=-tan^-1(oo)`

`=-tan^-1(tan  pi/2)`

`=pi/2`

`thereforetan^-1x+tan^-1  1/x=-pi/2, x<0`