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Question
Prove `tan^(-1) 1/5 + tan^(-1) (1/7) + tan^(-1) 1/3 + tan^(-1) 1/8 = pi/4`
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Solution
L.H.S = `tan^(-1) 1/5 + tan^(-1) 1/7 + tan^(-1) 1/3 + tan^(-1) 1/8`
= `tan^(-1) ((1/5 + 1/7)/(1-1/5 xx 1/7)) + tan^(-1) ((1/3 + 1/8)/(1-1/3 xx 1/8))` `" "[tan^(-1) x + tan^(-1) y = tan^(-1) (x + y)/(1 - xy)]`
`= tan^(-1) ((7+5)/(35-1)) + tan^(-1) ((8 + 3)/(24 - 1))`
`= tan^(-1) 12/34 + tan^(-1) 11/23`
= `tan^(-1) ((6/17 + 11/23 )/(1- 6/17 xx 11/23))`
`= tan^(-1) ((138 + 187)/(391 - 66))`
`= tan^(-1) (325/325) = tan^(-1) `
`= pi/4` = R.H.S
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