Explanation:

sin 10° sin 30° sin 50° sin 70° = (1/2) sin10° sin50° sin70°
Multiplying and dividing by 2cos10° we get,
∴ sin 10° sin 30° sin50° sin 70° = 2 × cos10° sin10° sin50° sin70° × (1/(4cos10°))
= sin20° sin50° sin70° × (1/(4cos10°))      ----(? sin2A = 2sinAcosA)
Multiply and divide by 2,
sin 10° sin 30° sin50° sin 70° = 2sin20° sin50° sin70° × (1/8cos10°)
= 2sin20° sin50° sin(90 - 20)° × (1/8cos10°)
= 2sin20° sin50° cos20° × (1/8cos10°)
= Sin40° sin50° × (1/8cos10°)      ----(? sin2A = 2sinAcosA)
Multiply and divide by 2,
= 2 × Sin40° sin(90 - 40)° × (1/16cos10°)
= 2 × Sin40° cos40° × (1/16cos(90 - 80)°)
= sin80°/(16sin80°) = (1/16)
∴ sin 10° sin 30° sin50° sin 70° = 1/16