Prove the following: `4tan^(-1)1/5-tan^(-1)1/(70)+tan^(-1)1/(99)=pi/4` `2t a n^(-1)1/5+s e c^(-1)(5sqrt(2))/7+2tan^(-1)1/8=pi/4`

Explanation

`4 tan ^{-1} frac{1}{5}-tan ^{-1} frac{1}{70}+tan ^{-1} frac{1}{99} `

`=2{2 tan ^{-1} frac{1}{5}}-{tan ^{-1} frac{1}{70}-tan ^{-1} frac{1}{99}} `

`=2{tan ^{-1}(frac{2 times 1 / 5}{1-(1 / 5)^{2}})}-tan ^{-1}{frac{frac{1}{70}-frac{1}{99}}{1+frac{1}{20} times frac{1}{99}}} `

`=2 tan ^{-1} frac{5}{12}{tan ^{-1}(frac{29}{6931})} `

`=tan ^{-1}{frac{2 times 5 / 12}{1-(5 / 12)^{2}}}-tan ^{-1}(frac{1}{239}) `

`=tan ^{-1} frac{120}{119}-tan ^{-1} frac{1}{239} `

`=tan ^{-1}{frac{frac{120}{119}-frac{1}{239}}{1+frac{1219}{119} times frac{1}{239}}} `

`=tan ^{-1} 1=frac{pi}{4}`