Cos-15cos2 -15cos4 -15cos7 -15-1-16

LHS = cos π/15 cos 2 π/15 cos 4 π/15 cos 7 π/15 = 2 sin π/15. cos π/15/2 sin π/15. cos 2 π/15. cos 4 π/15. cos 7 π/15 [Dividding and multiply by 2 in π/15] = 2 ...

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Byju's Answer

Standard XII

Mathematics

Product of Trigonometric Ratios in Terms of Their Sum

cosπ/15cos2 π...

Question

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

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Solution

$L H S = c o s \frac{π}{15} c o s \frac{2 π}{15} c o s \frac{4 π}{15} c o s \frac{7 π}{15} = \frac{2 s i n \frac{π}{15} . c o s \frac{π}{15}}{2 s i n \frac{π}{15}} . c o s \frac{2 π}{15} . c o s \frac{4 π}{15} . c o s \frac{7 π}{15}$

[Dividding and multiply by 2 in $\frac{π}{15}$ ]

$= \frac{2 s i n \frac{2 π}{15}}{2.2 s i n \frac{π}{15}} . c o s \frac{2 π}{15} . c o s \frac{4 π}{15} . c o s \frac{7 π}{15} = \frac{2 s i n \frac{4 π}{15}}{2.4 s i n \frac{π}{15}} . c o s \frac{4 π}{15} . c o s \frac{7 π}{15} = \frac{2 s i n \frac{8 π}{15}}{2.8 s i n \frac{π}{15}} . c o s (\frac{7 π}{15})$

$= \frac{s i n (\frac{8 π}{15} + \frac{7 π}{15}) + s i n (\frac{8 π}{15} - \frac{7 π}{15})}{16 s i n \frac{π}{15}} = \frac{s i n π + s i n \frac{π}{15}}{16 s i n \frac{π}{15}} [∵ s i n π = 0] = \frac{1}{16}$
= RHS

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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. 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}

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. 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 4π15 cos 7π15=116

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