Cos-15cos2 -15cos3 -15cos4 -15cos5 -15cos6 -15cos7 -15-1-128

Cos 5 π/15= cos π/3 = 1/2 cos 7 π/15= cos (π 8 π/15 ) cos 7 π/15= cos 8 π/15 Now LHS = cos π/15 cos 2 π/15 cos 4 π/15 cos 3 π/15 cos 5 π/15 cos 6 π/15 cos 7 π ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Multiples of an Angle

cosπ/15cos2 π...

Question

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

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Solution

$c o s \frac{5 π}{15} = c o s \frac{π}{3} = \frac{1}{2} c o s \frac{7 π}{15} = c o s (π - \frac{8 π}{15}) c o s \frac{7 π}{15} = - c o s \frac{8 π}{15} Now LHS = c o s \frac{π}{15} c o s \frac{2 π}{15} c o s \frac{4 π}{15} c o s \frac{3 π}{15} c o s \frac{5 π}{15} c o s \frac{6 π}{15} c o s \frac{7 π}{15} = [c o s \frac{π}{15} c o s \frac{2 π}{15} c o s \frac{4 π}{15} (- c o s \frac{8 π}{15})] (c o s \frac{3 π}{15} c o s \frac{6 π}{15}) \frac{1}{2}$

$= \frac{2^{3}}{2^{4} s i n \frac{π}{15}} . [2 s i n \frac{π}{15} c o s \frac{π}{15} c o s \frac{2 π}{15} c o s \frac{4 π}{15} c o s \frac{8 π}{15}] \times \frac{2}{8 s i n \frac{3 π}{15}} (2 s i n \frac{3 π}{15} c o s \frac{3 π}{15} c o s \frac{6 π}{15}) = - \frac{2^{3}}{16 s i n \frac{π}{15}} [s i n \frac{2 π}{15} c o s \frac{2 π}{15} c o s \frac{4 π}{15} c o s \frac{8 π}{15}] \times \frac{2}{8 s i n \frac{3 π}{15}} (s i n \frac{6 π}{15} c o s \frac{6 π}{15}) = \frac{2^{2}}{16 s i n \frac{π}{15}} [s i n \frac{2 π}{15} c o s \frac{2 π}{15} c o s \frac{4 π}{15} c o s \frac{8 π}{15}] \times \frac{1}{8 s i n \frac{3 π}{15}} (2 s i n \frac{6 π}{15} c o s \frac{6 π}{15})$

$= - \frac{2}{16 s i n \frac{π}{15}} [s i n \frac{8 π}{15} c o s \frac{8 π}{15}] \frac{s i n \frac{12 π}{15}}{8 s i n \frac{3 π}{15}} = - \frac{1}{16 s i n \frac{π}{15}} (s i n \frac{16 π}{15}) \frac{s i n \frac{12 π}{15}}{8 s i n \frac{3 π}{15}} = - \frac{- s i n (π + \frac{π}{15})}{128 s i n \frac{π}{15}} \times \frac{s i n (π - \frac{3 π}{15})}{s i n \frac{3 π}{15}} = - \frac{- s i n \frac{π}{15}}{128 s i n \frac{π}{15}} \times \frac{s i n \frac{3 π}{15}}{s i n \frac{3 π}{15}} = \frac{1}{128}$

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Similar questions

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

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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cos π15 cos 2π15 cos 3π15 cos 4π15 cos 5π15 cos 6π15 cos 7π15=1128

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