The value of the product (1 + tan1°)(1 + tan2°)(1 + tan3°) ⋯ (1 + tan44°)(1 + tan45°) is:

Whenever $A+B=45$° we have tanA + tanB = 1 − tanA·tanB, so (1 + tanA)(1 + tanB) = 1 + (tanA + tanB) + tanA·tanB = 2. Pair the brackets so the angles add to 45°: (1°, 44°), (2°, 43°), … , (22°, 23°) — 22 pairs, each contributing 2 — and the unpaired bracket is 1 + tan45° = 2. The product is $2^{22}\times 2=2^{23}=8388608$. Forgetting the leftover bracket gives $2^{22}$, and counting one factor of 2 for every bracket gives $2^{45}$.