The Arctan or Arctangent is the inverse of Tangent, which is a ratio, generally referring to the degrees of angles, when of course, x is a Real number.

Therefore, arctan x can be many things depending upon the value of x. Let’s explore some examples,

If,

tan y = x

Then the arctangent of x is equal to the inverse tangent function of x, which is equal to y:

arctan x= tan^-1 x = y

Arctan(x) can be represented in the form of a McLaurin series.

For Example,

Recall the formula for sum of a geometric series (with a common ratio rr such that |r|<1|r|<1) :

∑n = 0 ∞ φn = 11 −φ ∑n = 0 ∞ φn = 11−φ

Let φ=−t2φ=−t2.

⟹∑n = 0 ∞ (−t2) n = 11 + t2

⟹∑n = 0 ∞ (−t2) n = 11+ t2

(−t2) n (−t2) n can be rewritten as (−1) nt2n (−1) nt2n.

Hence,

∑n = 0 ∞ (−1) nt2n = 11 + t2∑n = 0 ∞ (−1) nt2n =11+ t2

Take the integral of both sides over [0, x] [0, x]:

∑n = 0 ∞ ∫0x (−1) nt2ndt = ∫0x11 + t2dt ∑n = 0 ∞ ∫0x (−1) nt2ndt = ∫0x11 + t2dt

The RHS is just a standard Integral of arctan(x)

arctan⁡(x):

Therefore,

∑n = 0 ∞ ∫0x (−1) nt2ndt = arctan(x)

Therefore,

∑n= 0 ∞ ∫0x (−1) nt2ndt = arctan⁡(x)

Integrating the LHS:

∑n = 0 ∞ (−1) n (t2n+12n+1∣∣∣x0) = arctan(x)

Therefore,

arctan(x) = ∑n= 0 ∞ (−1) nx2n + 12n + 1

Derivative of arctan(x):

Let’s use our formula for the derivative of an inverse function to find the derivative of the inverse of the tangent function:

y = tan−1

 x = arctan x.

We simplify the equation by taking the tangent of both sides:

y = tan−1 x

tan y = tan (tan−1 x)

Therefore,

tan y = x

Graphical Presentation of Arctan(x)