`tan^(-1).(1)/(3)+tan^(-1).(1)/(5)+tan^(-1).(1)/(7)+tan^(-1).(1)/(8)=???.`

Options

A. `(pi)/(4)`
B. `(pi)/(6)`
C. `(pi)/(8)`
D. `(pi)/(2)`

Answer

A

Explanation

`L.H.S.=tan^(-1).(1)/(3)+tan^(-1).(1)/(5)+tan^(-1).(1)/(7)+tan^(-1).(1)/(8)`
`=(tan^(-1).(1)/(3)+tan^(-1).(1)/(5))+(tan^(-1).(1)/(7)+tan^(-1).(1)/(8))`
`=tan^(-1)(((1)/(3)+(1)/(5))/(1-(1)/(3)xx(1)/(5)))+tan^(-1)(((1)/(7)+(1)/(8))/(1-(1)/(7)xx(1)/(8))),`
`[because tan^(-1)x+tan^(-1)y=tan^(-1)((x+y)/(1-xy)), " यदि " xy lt 1]`
`tan^(-1)(((8)/(15))/((14)/(15)))+tan^(-1)(((15)/(56))/((55)/(56)))`
`=tan^(-1)((8)/(14))+tan^(-1)((15)/(55))`
`=tan^(-1).(4)/(7)+tan^(-1).(3)/(11)`
`=tan^(-1)(((4)/(7)+(3)/(11))/(1-(4)/(7)xx(3)/(11)))=tan^(-1)(((44+21)/(77))/((77-12)/(77)))`
`=tan^(-1)(((65)/(77))/((65)/(77)))=tan^(-1)(1)`
`=(pi)/(4)=R.H.S." "`सिद्ध करना था।