Evaluate: tan5°·tan85° + tan15°·tan75° + tan25°·tan65° + tan35°·tan55° + tan45°·tan45° =

In each of the first four products the two angles add to 90°, so the second factor is the cotangent of the first: tan5°·tan85° = tan5°·cot5° = 1, and likewise for (15°, 75°), (25°, 65°) and (35°, 55°). The last term is tan45°·tan45° = $1\times 1=1$, since 45° is its own complement. Five terms, each worth 1, total 5. Overlooking the self-paired last term leaves 4, adding a half for it gives 9/2, and reading the five terms as one long product gives 1.