Inverse Tangent

Definition 1: Inverse Tangent

y = tan^-1x if and only if tan y = x where -π/2 is less than y is less than π/2.

Lemma: sec²x = 1 + tan² x

Proof:

(1) By definition sec x = 1/cos x

(2) So, sec²x = 1/(cos² x)

(3) sin²x + cos²x = 1 [See Corollary 2, here]

(4) 1/(cos² x) = (cos²x + sin²x)/(cos²) =

= cos²x/(cos²x) + (sin²x)/(cos²) = 1 + tan²x.

QED

Theorem: D tan^-1 x = 1/(1 + x²)

Proof:

(1) Let y = tan^-1 x

(2) Then tan y = x [See Definition 1 above]

(3) d(tan y)/dx = d(x)/dx = 1

(4) d(tan y)/dx = (sec²y)dy/dx = 1 [See Theorem 1, here]

(5) dy/dx = 1/sec²y

(6) Using Lemma 1 above, we have:

dy/dx =1/(1 + tan²y)

(6) Using step #2, this gives us:

dy/dx = 1/(1 + tan²y) = 1/(1 + x²) [Since x = tan y]

QED

References

• Edwards & Penny, Calculus and Analytic Geometry