Example 4: Arc Tangent

Find cos (arctan y). Let θ = arctan y. Thus tan θ = y. Using

we solve for cos θ.

sin θ = y cos θ,              (y cos θ)² + cos² θ=1,

cos² θ(y² + 1) = 1,

Thus

By definition of arctan y, we know that -π/2 ≤ θ ≤ π/2. In this interval, cos θ ≥ 0. Therefore