Question

Prove the following: `2 tan^(-1)3/4-tan^(-1)(17)/(31)=tan^(-1)pi/4`

Answer 1

We have,
LHS `=tan^(-1)3/4-tan17/31`
`=tan^(-1){(2 xx 3/4)/(1-(3/4)^(2)}-tan^(-1)17/31`
`=tan^(-1)24/7-tan^(-1)17/31=tan^(-1){(24/7-17/31)/(1+(24/7 xx 17/31)}`
`=tan^(-1){625/217 xx 217/625} = tan^(-1) 1 = pi/4` = RHS.
Hence, `2tan^(-1)3/4-tan^(-1)17/31=pi/4`.