Question

Evaluate:

`sin(tan^-1x+tan^-1  1/x)` for x < 0

Answer 1

`sin(tan^-1x+tan^-1  1/x)=sin[tan^-1(-x)+tan^-1(-1/x)]`    `[thereforex<0]`

`=sin[-tan^-1(x)-tan^-1(1/x)]`

`=sin{-[tan^-1(x)+tan^-1(1/x)]}`

`=sin[-(tan^-1x+cot^-1x)]`       `[thereforetan^-1  1/x=cot^-1x]`

`=-sin(tan^-1x+cot^-1x)`

`=-sin(pi/2)`    `[thereforetan^-1x+cot^-1x=pi/2]`

= -1