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Question
Find the value of the given expression.
`tan^(-1) (tan (3pi)/4)`
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Solution
We know that tan–1 (tan x) = x if `x ∈ (-pi/2, pi/2)`, which is the principal value branch of tan–1 x.
Here, `(3pi)/4 ∉ ((-pi)/2, pi/2)`.
Now, `tan^(-1) (tan (3pi)/4)` can be written 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 ∈ ((-pi)/2, pi/2)`
∴ `tan^(-1) (tan (3pi)/4)`
= `tan^(-1) [tan((-pi)/4)]`
= `(-pi)/4`
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