2020 · Q53
Let 0° < θ < 90° and 100θ = 90°. If α = Π_(n=1)^(99) cot(nθ), then which one of the following is correct?
Elementary MathematicsTrigonometry
- Aα = 1
- Bα = 0
- Cα > 1
- D0 < α < 1
Explanation
100θ = 90° gives θ = 0.9°. Pair the terms n and 100 − n: cot(nθ)·cot((100−n)θ) = cot(nθ)·tan(nθ) = 1. There are 49 such pairs plus the middle term cot(50θ) = cot45° = 1, so α = 1.