Let's simplify the expression step by step:

tg 7α - tg 2α ÷ 1 + tg 7α · tg 2α
= (tg 7α - tg 2α) / (1 + tg 7α · tg 2α)

We know that tg(a) - tg(b) = sin(a)cos(b) - cos(a)sin(b) = sin(a-b)
Therefore, tg 7α - tg 2α = sin(7α - 2α) = sin(5α)

Similarly, we know that tg(a) · tg(b) = sin(a)cos(a) / cos(b)sin(b) = sin(2a)cos(2b)
Therefore, tg 7α · tg 2α = sin(14α)cos(4α)

Now, let's simplify the expression further:

= (sin(5α)) / (1 + sin(14α)cos(4α))

This is the simplified form of the given expression: (sin(5α)) / (1 + sin(14α)cos(4α))