Question

16sin10sin30sin50sin70=1

Answer 1

sin10sin30sin50sin70
sin30sin50sin70sin10
1/2(sin70sin50)sin10
multiply by 2 in numerator and denominator
1/4(2sin70sin50)sin10
1/4(sin120sin20)sin10
(sin120=1/2)
1/4(-sin10+2sin20sin10)
1/8(-sin10+sin30+sin10)
1/8(sin30)
1/8(1/2)
1/16 hence proved

Answer 2

$16sin10sin30sin50sin70 \\ \\ = 16 \times (\frac{1}{2} sin10sin50sin70) \\ \\ = 16 \times \frac{1}{2} \times \frac{1}{2} (2sin10sin50)sin70 \\ \\ = 4(cos(10 - 50) - cos(10 + 50))sin70 \\ \\ = 4(cos40 - cos60)sin70 \\ \\ = 4(cos40sin70 - \frac{1}{2} sin70) \\ \\ = \frac{4}{2} (2cos40sin70 - sin70) \\ \\ = 2(sin(40 + 70) - sin(40 - 70) - sin70) \\ \\ = 2(sin110 + sin30 - sin70) \\ \\ = 2(sin(90 + 20) + sin30 - sin(90 - 20)) \\ \\ = 2(cos20 + \frac{1}{2} - cos20) \\ \\ = 2 \times \frac{1}{2} \\ \\ = 1$