Question

prove that sin 10 sin 30 sin 50 sin 70 = 1/16
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Answer 1

$\huge{\dag} \: {\underline{\boxed{\sf{\purple{ Sᴏʟᴜᴛɪᴏɴ \ :-}}}}}$

We Should Know Some Trignometric Identities For Solving This Question.

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$1. \: \: \sin(2 \alpha ) = 2 \sin( \alpha ) \cos( \alpha )$

$2. \: \: \sin(180 - \alpha ) = \sin( \alpha )$

$3. \: \: \sin(90 - \alpha ) = \cos( \alpha )$

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L.H.S :- Sin10. Sin30. Sin50. Sin70

$= \cos(90 - 10) \sin(30) \cos(90 - 40) \cos(90 - 70) \\ = \cos(80) \: \frac{1}{2} \: \cos(40) \: \cos(20) \\ = \frac{1}{4 \sin(20) } \cos(8 0 ) \cos(40) . \: 2 \sin(20) \cos(20) \\ = \frac{1}{8 \sin(20) } \cos(80) 2. \cos(40) \sin(40) \\ = \frac{1}{16 \sin(20) } 2 \cos(80) \sin(80 ) \\ = \frac{1}{16 \sin(20) } \sin(160) \\ = \frac{1}{16 \sin(20) } \sin(180 - 20) \\ = \frac{1}{16 \sin(20) } \sin(20) \\ = \frac{1}{16} \: \: \: \: \: \: \: \: .........R.H.S$