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Prove that ta n^-1 1/3 + ta n ^-1 1/5 +...

Prove that `ta n^-1 1/3 + ta n ^-1 1/5 + ta n ^-1 1/7 + ta n ^-1 1/8 = pi/4`

A

`(11pi)/(5)`

B

`(pi)/(4)`

C

`pi`

D

`(3pi)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
B
We have,
`"tan"^(-1)(1)/(3)+"tan"^(-1)(1)/(5)+"tan"^(-1)(1)/(7)+"tan"^(-1)(1)/(8)`
`implies tan^(-1)(((1)/(3)+(1)/(5))/(1-(1)/(3)xx(1)/(5)))+"tan"^(-1)(((1)/(7)+(1)/(8))/(1-(1)/(7)xx(1)/(8)))`
`(becausetan^(-1)x+tan^(-1)y=tan^(-1)((x+y)/(1-xy)))`
`"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)/(77-12))`
`=tan^(-1)((65)/(65))=tan^(-1)(1)=(pi)/(4)`
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