Question 71219

Concept: for any x, cot(cot-1x) = x, so tan(cot-1x) = 1/x whenever x is not 0. For an angle θ with sin θ not 0, the half-angle identity gives cosec θ - cot θ = (1 - cos θ)/sin θ = tan(θ/2). And tan-1(tan u) = u holds exactly when u lies in (-π/2, π/2).

Application: put θ = tan-1x with x not 0, so that tan θ = x and θ lies in (-π/2, π/2).

1. From θ = tan-1x we get tan θ = x, hence cot θ = 1/x and cosec θ = √(1 + x2)/x.

2. Because cot(cot-1x) = x, the second term inside the bracket is tan(cot-1x) = 1/x.

3. So the bracket is cosec θ - cot θ = √(1 + x2)/x - 1/x = (√(1 + x2) - 1)/x.

4. Written in terms of θ, cosec θ - cot θ = (1 - cos θ)/sin θ = tan(θ/2), so the bracket equals tan(θ/2).

5. Since θ lies in (-π/2, π/2), θ/2 lies in (-π/4, π/4), which is inside the principal branch, so tan-1(tan(θ/2)) = θ/2.

6. Therefore 2tan-1[cosec(tan-1x) - tan(cot-1x)] = 2 × (θ/2) = θ = tan-1x.

Cross-check: take x = 1.

1. Then cosec(tan-11) = cosec(π/4) = √2 and tan(cot-11) = tan(π/4) = 1, so the bracket is √2 - 1, about 0.4142.

2. Now 2tan-1(0.4142) is about 2 × 0.3927 = 0.7854, which is π/4, exactly the value of tan-11.

Contrast: evaluating the four candidate expressions at x = 1 separates them by value.

The inverse cosecant is in addition undefined for |x| < 1, while the given expression is defined for every x other than 0. The value π/4, about 0.7854, is exactly what the derivation and the numerical check produced.

Result: the given expression equals tan-1x for every x other than 0.