Question

How do you evaluate `2((tan^-1(1/3))-tan^-1(-1/7)`?

Answer 1

`pi/4`.

Explanation:

We have to use these Rules : `R(1) : tan^-1x+tan^-1y=tan^-1{(x+y)/(1-xy)}; x,y>0, xy<1`.

`R(2) : tan^-1(-x)=-(tan^-1x), x in RR`.

The Given Exp. `=2tan^-1(1/3)-tan^-1(-1/7)`
`=tan^-1(1/3)+tan^-1(1/3)+tan^-1(1/7).................[R(2)]`

`=tan^-1{(1/3+1/3)/(1-1/3*1/3)}+tan^-1(1/7)................[R(1)]`

`=tan^-1(2/3*9/8)+tan^-1(1/7)`
`tan^-1(3/4)+ tan^-1(1/7)`
`=tan^-1{(3/4+1/7)/(1-3/28)}................{R(1)]`
`=tan^-1{(25/28)/(25/28)}`
`=tan^-1(1)`
`=pi/4`.