Prove that sin 10° sin 30° sin 50° sin 70° = \(\frac{1}{16}\)

1.	Prove that sin 10° sin 30° sin 50° sin 70° = \(\frac{1}{16}\)
Answer» L.H.S = sin 10° sin 30° sin 50° sin 70° = \(\frac{1}{2}\) (sin 10° sin 50° sin 70° ) = \(\frac{1}{2}\) (sin 50° sin 70°) sin 10° = \(\frac{1}{2}\) [sin (60° − 10°) sin (60° + 10°)] sin 10° = \(\frac{1}{2}\) [(\(sin^260°-sin^210°\))] sin 10° = \(\frac{1}{2}\)(( \(\frac{3}{2}\))² − \(sin^210°\)) sin 10° = \(\frac{1}{2}\) [ \(\frac{3}{4}\) − \(sin^210°\)] sin 10° = \(\frac{1}{8}\) (3 − 4 \(sin^210°\))sin 10° = \(\frac{1}{8}\)(3 sin 10° − 4 \(sin^210°\)) = \(\frac{1}{11}\) (sin 3 × 10°) = \(\frac{1}{8}\) (sin 30°) = \(\frac{1}{16}\) = R.H.S

Answer»

L.H.S

= sin 10° sin 30° sin 50° sin 70°

= \(\frac{1}{2}\) (sin 10° sin 50° sin 70° )

= \(\frac{1}{2}\) (sin 50° sin 70°) sin 10°

= \(\frac{1}{2}\) [sin (60° − 10°) sin (60° + 10°)] sin 10°

= \(\frac{1}{2}\) [(\(sin^260°-sin^210°\))] sin 10°

= \(\frac{1}{2}\)(( \(\frac{3}{2}\))² − \(sin^210°\)) sin 10°

= \(\frac{1}{2}\) [ \(\frac{3}{4}\) − \(sin^210°\)] sin 10°

= \(\frac{1}{8}\) (3 − 4 \(sin^210°\))sin 10°

= \(\frac{1}{8}\)(3 sin 10° − 4 \(sin^210°\))

= \(\frac{1}{11}\) (sin 3 × 10°)

= \(\frac{1}{8}\) (sin 30°)

= \(\frac{1}{16}\) = R.H.S

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