Question

If sin²(10°)sin(20°)sin(40°)sin(50°)sin(70°) 
\(=α−\frac{1}{16}sin⁡(10^∘),\)
 then 16 + α^–1 is equal to _______

Answer 1

Correct Answer: 80

The correct answer is 80
(sin10° ⋅ sin50° ⋅ sin70°).(sin10° ⋅ sin20° ⋅ sin40°)
\(=(\frac{1}{4}sin⁡30^∘)⋅[\frac{1}{2}sin⁡10^∘(cos⁡20^∘−cos⁡60^∘)]\)
\(=\frac{1}{16}[sin⁡10^∘(cos⁡20^∘−\frac{1}{2})]\)
\(=\frac{1}{32}[2sin⁡10^∘⋅cos⁡20^∘−sin⁡10^∘]\)
\(=\frac{1}{32}[sin⁡30^∘−sin⁡10^∘−sin⁡10^∘]\)
\(=\frac{1}{64}−\frac{1}{16}sin⁡10^∘\)
Clearly \(α=\frac{1}{64}\)
Therefore,
16 + α^–1 = 80