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Question
Find the value of `4tan^-1 1/5 - tan^-1 1/239`
Solution
`4tan^-1 1/5 - tan^-1 1/239`
= `2(2tan^-1 1/5) - tan^-1 1/239`
= `2tan^-1 (2/5)/(1 - (1/5)^2) - tan^-1 1/239` .....`(because 2tan^-1x = tan^-1 (2x)/(1 - x^2))`
= `2tan^-1 (2/5)/(24/25) - tan^-1 1/239`
= `2tan^-1 5/12 - tan^-1 1/239`
= `2tan^-1 (2/5)/(1 - (1/5)^2) - tan^-1 1/239` .....`(because 2tan^-1x = tan^-1 (2x)/(1 - x^2))`
= `2tan^-1 (2/5)/(24/25) - tan^-1 1/239`
= `2tan^-1 5/12 - tan^-1 1/239`
= `tan^-1 (2*5/12)/(1 - (5/12)^2) - tan^-1 1/239` ......`(because 2tan^-1x = tan^-1 (2x)/(1 - x^2))`
= `tan^-1 (144 xx 5)/(119 xx 6) - tan^-1 1/239`
= `tan^-1 120/119 - tan^-1 1/239`
= `tan^-1 (120/119 - 1/239)/(1 + 120/119 * 1/239)` ......`(because tan^-1x - tan^-1y = tan^-1 (x - y)/(1 + xy))`
= `tan^-1 (120 xx 239 - 119)/(119 xx 239 + 120)`
= `tan^-1 (28680 - 119)/(28441 + 120)`
= `tan^-1 28561/28561`
= `tan^-1 1 = pi/4`
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