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Question Number 91465 by Zainal Arifin last updated on 30/Apr/20

Commented by Prithwish Sen 1 last updated on 01/May/20

8.((2.sin((2π)/(15)).cos((2π)/(15)))/(sin((2π)/(15)))) .cos((4π)/(15))cos((8π)/(15))cos((14π)/(15)) =2.((sin((16π)/(15)))/(sin((2π)/(15)))).cos((14π)/(15))=2.((sin(π+(π/(15))))/(sin((2π)/(15)))).cos(π−(π/(15))) =2.((sin(π/(15)))/(2.sin(π/(15)).cos(π/(15)))) . cos(π/(15)) = 1

8.2.sin2π15.cos2π15sin2π15.cos4π15cos8π15cos14π15=2.sin16π15sin2π15.cos14π15=2.sin(π+π15)sin2π15.cos(π−π15)=2.sinπ152.sinπ15.cosπ15.cosπ15=1

Answered by john santu last updated on 01/May/20

let (π/(15)) = w , cos ((14π)/(15)) = cos(π−(π/(15))) = −cos (π/(15)) ⇒ ((−16cos w cos 2w cos 4w cos 8w)/(sin w)) × sin w = ((−8 sin 2w cos 2w cos 4w cos 8w)/(sin w)) = ((−4sin 4w cos 4w cos 8w)/(sin w)) = ((−2sin 8w cos 8w)/(sin w)) = ((−sin 16w)/(sin w)) sin (((16π)/(15))) = sin (π+(π/(15))) = −sin ((π/(15))) ∴ ((sin ((π/(15))))/(sin ((π/(15))))) = 1

letπ15=w,cos14π15=cos(π−π15)=−cosπ15⇒−16coswcos2wcos4wcos8wsinw×sinw=−8sin2wcos2wcos4wcos8wsinw=−4sin4wcos4wcos8wsinw=−2sin8wcos8wsinw=−sin16wsinwsin(16π15)=sin(π+π15)=−sin(π15)∴sin(π15)sin(π15)=1

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