Prove that cos- 15 cos 2- 15 cos 3- 15 cos 4- 15 cos 5- 15 cos 6- 15 cos 7- 15 - 127

LHS=cosπ #160; 15 #160; cos 2π #160; 15 #160; cos 3π #160; 15 #160; cos 4π #160; 15 #160; cos 5π #160; 15 #160; cos 6π #160 ...

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Byju's Answer

Standard XII

Mathematics

Integration by Substitution

Prove that c...

Question

Prove that
$cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{3 π}{15} cos \frac{4 π}{15} cos \frac{5 π}{15} cos \frac{6 π}{15} cos \frac{7 π}{15} = \frac{1}{2^{7}}$

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Solution

$L H S = cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{3 π}{15} cos \frac{4 π}{15} cos \frac{5 π}{15} cos \frac{6 π}{15} cos \frac{7 π}{15} = cos \frac{π}{15} cos \frac{2 π}{15} cos (π - \frac{12 π}{15}) cos \frac{4 π}{15} cos \frac{π}{3} cos \frac{6 π}{15} cos (π - \frac{8 π}{15}) = (cos \frac{π}{15} . cos \frac{2 π}{15} cos \frac{4 π}{15} cos (π - \frac{8 π}{15})) (cos \frac{π}{3}) (cos (\frac{6 π}{15}) . cos (\frac{8 π}{15})) = \frac{- 1}{2^{4}} . \frac{1}{2} - \frac{1}{2^{2}} = \frac{1}{2^{7}}$

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Similar questions

$c o s \frac{π}{15} c o s \frac{2 π}{15} c o s \frac{3 π}{15} c o s \frac{4 π}{15} c o s \frac{5 π}{15} c o s \frac{6 π}{15} c o s \frac{7 π}{15} = \frac{1}{128}$

Q. The value of

cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{3 π}{15} cos \frac{4 π}{15} cos \frac{5 π}{15} cos \frac{6 π}{15} cos \frac{7 π}{15}

is equal to

Q. Show that

cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{3 π}{15} cos \frac{4 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15} cos \frac{7 π}{15} = \frac{1}{128}

$c o s \frac{π}{15} c o s \frac{2 π}{15} c o s \frac{4 π}{15} c o s \frac{7 π}{15} = \frac{1}{16}$

Q. The value of

cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{8 π}{15}

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Prove thatcosπ15cos2π15cos3π15cos4π15cos5π15cos6π15cos7π15=127

LHS=cosπ15cos2π15cos3π15cos4π15cos5π15cos6π15cos7π15=cosπ15cos2π15cos(π−12π15)cos4π15cosπ3cos6π15cos(π−8π15)=(cosπ15.cos2π15cos4π15cos(π−8π15))(cosπ3)(cos(6π15).cos(8π15))=−124.12−122=127

Prove that
$cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{3 π}{15} cos \frac{4 π}{15} cos \frac{5 π}{15} cos \frac{6 π}{15} cos \frac{7 π}{15} = \frac{1}{2^{7}}$