Recall that

So, letting θ = α + β, and expanding using our new sine and cosine identities, we have:

Dividing numerator and denominator by cos α cos β:

Simplifying gives us:

Replacing β with (-β) gives us

[The tangent function is odd, so tan(-β) = - tan β]

We have proved the two tangent of the sum and difference of two angles identities: