Prove the double angle formula for tangent.

The double angle formula for tangent is tan(2θ) = (2tanθ)/(1-tan²θ).

To prove the double angle formula for tangent, we start with the identity for tangent of the sum of two angles:

tan(α + β) = (tanα + tanβ)/(1 - tanαtanβ)

Let α = θ and β = θ, so we have:

tan(2θ) = (tanθ + tanθ)/(1 - tanθtanθ)
tan(2θ) = 2tanθ/(1 - tan²θ)

Therefore, we have proven the double angle formula for tangent.