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Question
Find the value of the given expression.
`tan^(-1) (tan (3pi)/4)`
Solution
`tan^(-1) (tan (3pi)/4)`
We know that tan−1 (tan x) = x if x in `(-pi/2, pi/2)` which is the principal value branch of tan−1x.
Here `(3pi)/4 !in ((-pi)/2, pi/2)`
Now , `tan^(-1) (tan (3pi)/4)` can be witten as
`tan^(-1) (tan (3pi)/4) `
`= tan^(-1) [-tan ((-3pi)/4)]`
` = tan^(-1) [-tan(pi - pi/4)]`
`= tan^(-1) [-tan pi/4] `
`= tan^(-1) [tan(-pi/4)] " where " - pi/4 in ((-pi)/2, pi/2)`
`:. tan^(-1) (tan (3pi)/4)`
` = tan^(-1) [tan((-pi)/4)]`
` = (-pi)/4`
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