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Prove that 2tan^(-1)(1/5)+sec^(-1)((5√2)/7)+2tan^(-1)(1/8)=pi/4 - Mathematics

Prove that `2tan^(-1)(1/5)+sec^(-1)((5sqrt2)/7)+2tan^(-1)(1/8)=pi/4`

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Solution

`2tan^(-1)(1/5)+sec^(-1)((5sqrt2)/7)+2tan^(-1)(1/8)`

`=2tan^(-1)(1/5)+tan^(-1)(sqrt(((5sqrt2)/7)^2-1))+2tan^(-1)(1/8)  [`

`=2tan^(-1)(1/5)+tan^(-1)(1/7)+2tan^(-1)(1/8)`

`=2(tan^(-1)(1/5)+tan^(-1)(1/8))+tan^(-1)(1/7)`

`=tan^(-1)((1/5+1/8)/(1-(1/5)xx(1/8)))+tan^(-1)(1/7) [`

 

`=2 tan^(−1)(13/39)+tan^(−1)(1/7)`

 

`=2 tan^(−1)(1/3)+tan^(−1)(1/7)`

`= tan^(-1)((2/3)/(1-1/9))+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)(1)`

`=pi/4`

`=RHS`

Hence proved.

 

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