Question

Find the value of the given expression.

`tan^(-1) (tan  (3pi)/4)`

Answer 1

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`