सिद्ध कीजिए -
`2tan^(-1)((1)/(5))+sec^(-1)((5sqrt2)/(7))+2tan^(-1).(1)/(8)=(pi)/(4)`

Explanation

L.H.S.
`=2tan^(-1)((1)/(5))+sec^(-1)((5sqrt2)/(7))+2tan^(-1)((1)/(8))`
`=2[tan^(-1).(1)/(5)+tan^(-1).(1)/(8)]+tan^(-1)sqrt(((5sqrt2)/(7))^(3)-1),`
`" "[because sec^(-1)x=tan^(-1)sqrt(x^(2)-1)]`
`=2tan^(-1)(((1)/(5)+(1)/(8))/(1-(1)/(5)xx(1)/(8)))+tan^(-1)sqrt((50)/(49)-1),`
`" "[because tan^(-1)x+tan^(-1)y=tan^(-1).(x+y)/(1-xy)]`
`=2tan^(-1).(13)/(39)+tan^(-1).(1)/(7)`
`=2tan^(-1).(1)/(3)+tan^(-1).(1)/(7)`
`=tan^(-1)((2xx(1)/(3))/(1-((1)/(3))^(2)))+tan^(-1).(1)/(7),`
`" "[because 2tan^(-1)x=tan^(-1)((2x)/(1-x^(2)))]`
`=tan^(-1)(((2)/(3))/((8)/(9)))+tan^(-1).(1)/(7)`
`=tan^(-1)((3)/(4))+tan^(-1).(1)/(7)`
`=tan^(-1)(((3)/(4)+(1)/(7))/(1-(3)/(4)xx(1)/(7)))`
`=tan^(-1)((25)/(25))`
`=tan^(-1)(1)`
`=tan^(-1)(tan.(pi)/(4))=(pi)/(4)" "[because tan^(-1)(Tan theta)=theta]`
= R.H.S.
यही सिद्ध करना था।