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Prove that :2tan^(-1)1/5+sec^(-1)(5sqrt(...

Prove that :`2tan^(-1)1/5+sec^(-1)(5sqrt(2))/7+2tan^(-1)\ 1/8=pi/4`

A

`pi`

B

`(pi)/(3)`

C

`(pi)/(4)`

D

`(pi)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
`2tan^(-1)((1)/(5))+sec^(-1)((5sqrt(2))/(7))+2tan^(-1)((1)/(8))`
`=2("tan"^(-1)(1)/(5)+"tan"^(-1)(1)/(8))+sec^(-1)((5sqrt(2))/(7))`
`=tan^(-1)[((1)/(5)+(1)/(8))/(1-(1)/(5)xx(1)/(8))]+tan^(-1)sqrt(((5sqrt(2))/(7))^(2)-1)`
`["put"sec^(-1)x=thetarArrx=sectheta]`
Then , `tantheta=sqrt(sec^(2)theta-1)`
`rArrtheta=tan^(-1)sqrt(x^(2)-1)`
`rArrsec^(-1)x=tan^(-1)sqrt(x^(2)-1)]`
`=2"tan"^(-1)(13)/(39)+tan^(-1)sqrt((50)/(49))-1`
`=2"tan"^(-1)(1)/(3)+"tan"^(-1)(1)/(7)="tan"^(-1){(2xx(1)/(3))/(1-((1)/(3))^(2))}+"tan"^(-1)(1)/(7)" " [because2tan^(-1)x=tan^(-1){(2x)/(1-x^(2))}]`
`="tan"^(-1)(3)/(4)+"tan"^(-1)(1)/(7)=tan^(-1)(((3)/(4)+(1)/(7))/(1-((3)/(4)xx(1)/(7))))`
`=tan^(-1)((21+4)/(28-3))=tan^(-1)(1)=tan^(-1)("tan"(pi)/(4))=(pi)/(4)`
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