Question

How do you find the exact value of arctan(tanx)?

Answer 1

`arctan(tan x) = x - pi floor((x + pi/2)/pi)`

which simplifies to

`arctan(tan x) = x` if `x in (-pi/2, pi/2)`

Explanation:

If `x in (-pi/2, pi/2)` then `arctan(tan x) = x`

Otherwise we need to add some integer multiple of `pi` to `x` to bring it into this range.

Using the floor function, we can write:

`arctan(tan x) = x - pi floor((x + pi/2)/pi)`

Here's a graph of `arctan(tan(x))` :

graph{3pi/5(abs(sin(x/2+pi/4))-abs(cos(x/2+pi/4))-1/6(abs(sin(x/2+pi/4)^3))+1/6(abs(cos(x/2+pi/4)^3)))(tan(x/2+pi/4)/abs(tan(x/2+pi/4))) [-5, 5, -2.5, 2.5]}

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