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Question
Simplify: `tan^-1 x/y - tan^-1 (x - y)/(x + y)`
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Solution
`tan^-1 x/y - tan^-1 (x - y)/(x + y) = tan^-1 [(x/y - (x - y)/(x + y))/(1 + (x/y)((x - y)/(x + y)))]`
= `tan^-1 [((x(x + y) - y(x - y))/(y(x + y)))/(1 + (x(x - y))/(y(x + y)))]`
= `tan^-1 [(((x^2 + xy - xy + y^2))/(y(x + y)))/((y(x + y) + x(x - y))/(y(x + y)))]`
= `tan^-1 [(x^2 + y^2)/(xy + y^2 + x^2 - xy)]`
= `tan^-1 [(x^2 + y^2)/(x^2 + y^2)]`
= `tan^-1(1) = pi/4`
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