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

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