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