Math, asked by PranshuDewangan, 5 hours ago

Prove that sin 12° sin 40° sin 54°=1/8​

Answers

Answered by Starrex
9

\huge\sf\underline{To\:prove: }

ㅤㅤ\sf{\implies sin12°\:sin40°\:sin54°=\dfrac{1}{8}}

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\huge\sf\underline{Solution: }

Solving LHS :

ㅤㅤ\sf{\longrightarrow (sin12°\:sin40°)\:sin54°}

ㅤㅤ\sf{\longrightarrow \dfrac{1}{2}[(cos(12-48)-cos(12+48)]sin(90°-36°)}

ㅤㅤ\sf{\longrightarrow \dfrac{1}{2}(cos36°-cos60°)cos36°}

ㅤㅤ\sf{\longrightarrow \dfrac{1}{2}\left(\dfrac{\sqrt{5}+1}{4}-\dfrac{1}{2}\right)\left(\dfrac{\sqrt{5}+1}{4}\right)}

ㅤㅤ\sf{\longrightarrow \dfrac{1}{2\times 4\times 4}(\sqrt{5}+1-2)(\sqrt{5}+1)}

ㅤㅤ\sf{\longrightarrow \dfrac{1}{32}(\sqrt{5}-1)(\sqrt{5}+1)}

ㅤㅤ\sf{\longrightarrow \dfrac{1}{32}(5-1)}

ㅤㅤ\sf{\longrightarrow \cancel{\dfrac{4}{32}}}

ㅤㅤ\sf{\longrightarrow \dfrac{1}{8}}

ㅤㅤㅤㅤㅤㅤ\underline{\boxed{\bf{ Hence,\:proved!!}}}

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\large\sf\purple{\underline{Some\: identities\:used: }}

\sf{\implies sinA\:sinB= cos(A-B)-cos(A+B)}

\sf{\implies  cos(-x) = cos \:x}

\sf{\implies  sin(90°-x)=cos\:x}

\sf{\implies  cos36°=\dfrac{\sqrt{5}+1}{4}}

\sf{\implies  sin36°=\dfrac{\sqrt{5}-1}{4}}

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