Show that cos-15cos2-15cos3-15cos4-15cos5-15cos66-15cos7-15 - 1128

LHS=cosπ15cos2π15cos3π15cos4π15cos5π15cos66π15cos7π15 =cosπ15cos2π15cos4π15cos3π15cos5π15cos66π15cos7π15 =12sinπ15× 2sinπ15cosπ15cos2π15cos4π15co ...

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Byju's Answer

Standard VII

Mathematics

Area of a Triangle

Show that c...

Question

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

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Solution

$L H S = cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{3 π}{15} cos \frac{4 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15} cos \frac{7 π}{15}$
$= cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{3 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15} cos \frac{7 π}{15}$
$= \frac{1}{2 sin \frac{π}{15}} \times 2 sin \frac{π}{15} cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{3 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15} cos \frac{7 π}{15}$
$= \frac{1}{2 sin \frac{π}{15}} \times sin \frac{2 π}{15} cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{3 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15} cos \frac{7 π}{15}$
$= \frac{1}{4 sin \frac{π}{15}} \times 2 sin \frac{2 π}{15} cos \frac{2 π}{15} cos \frac{4 π}{15} cos \frac{3 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15} cos \frac{7 π}{15}$
$= \frac{1}{4 sin \frac{π}{15}} \times sin \frac{4 π}{15} cos \frac{4 π}{15} cos \frac{3 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15} cos \frac{7 π}{15}$
$= \frac{1}{8 sin \frac{π}{15}} \times 2 sin \frac{4 π}{15} cos \frac{4 π}{15} cos \frac{3 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15} cos \frac{7 π}{15}$
$= \frac{1}{8 sin \frac{π}{15}} \times sin \frac{8 π}{15} cos \frac{3 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15} cos (π - \frac{8 π}{15})$
$= \frac{- 1}{16 sin \frac{π}{15}} \times 2 sin \frac{8 π}{15} cos \frac{8 π}{15} cos \frac{3 π}{15} cos \frac{5 π}{15} cos \frac{66 π}{15}$
$= \frac{- 1}{16 sin \frac{π}{15}} \times sin \frac{16 π}{15} cos \frac{3 π}{15} cos \frac{π}{3} cos \frac{66 π}{15}$
$= \frac{- 1}{32 sin \frac{π}{15}} \times sin (π + \frac{π}{15}) cos \frac{3 π}{15} cos (4 π - \frac{6 π}{15})$
$= \frac{1}{32} \times cos \frac{π}{5} cos \frac{2 π}{5}$
$= \frac{1}{32} \times cos \frac{π}{5} cos \frac{2 π}{5}$
$= \frac{1}{32} \times cos 36^{\circ} cos (90^{\circ} - 18^{\circ})$
$= \frac{1}{32} \times \frac{\sqrt{5} + 1}{4} \times sin 18^{\circ}$
$= \frac{1}{32} \times \frac{\sqrt{5} + 1}{4} \times \frac{\sqrt{5} - 1}{4}$
$= \frac{1}{32} \times \frac{5 - 1}{16}$
$= \frac{1}{32} \times \frac{4}{16}$
$= \frac{1}{32} \times \frac{1}{4}$
$= \frac{1}{128} = R H S$
Hence proved.

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Similar questions

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

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}}

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

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

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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Show that cosπ15cos2π15cos3π15cos4π15cos5π15cos66π15cos7π15=1128

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