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Question
Solve `tan^(-1) - tan^(-1) (x - y)/(x+y)` is equal to
(A) `pi/2`
(B). `pi/3`
(C) `pi/4`
(D) `(-3pi)/4`
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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) y - tan^(-1) y tan^(-1) (x-y)/(1+ xy)] `
`= tan^(-1) [((x(x+y)-y(x-y))/(y(x+y)))/((y(x+y)+x(x-y))/(y(x+y)))]`
`= tan^(-1) ((x^2 + xy - xy + y^2)/(xy + y^2 + x^2 - xy))`
=` tan^(-1) ((x^2 + y^2)/(x^2 + y^2)) = tan^(-1) 1 = pi/4 `
Hence, the correct answer is C.
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