Cos-65cos2 -65cos4 -65cos8 -65cos16 -65cos32 -65-1-64

LHS=cos π/65 cos 2π/65 cos 4π/65 cos 8π/65 cos 16π/65 cos 32π/65 Divide and Multiply by 2 sin π/65, we get =2.sin π/65/2 sin π/65.cos π/65 cos 2π/65 cos 4π/65 c ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Multiples of an Angle

cosπ/65cos2 π...

Question

$c o s \frac{π}{65} c o s \frac{2 π}{65} c o s \frac{4 π}{65} c o s \frac{8 π}{65} c o s \frac{16 π}{65} c o s \frac{32 π}{65} = \frac{1}{64}$

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Solution

$L H S = c o s \frac{π}{65} c o s \frac{2 π}{65} c o s \frac{4 π}{65} c o s \frac{8 π}{65} c o s \frac{16 π}{65} c o s \frac{32 π}{65}$

Divide and Multiply by 2 sin $\frac{π}{65}$ , we get

$= \frac{2. s i n \frac{π}{65}}{2 s i n \frac{π}{65}} . c o s \frac{π}{65} c o s \frac{2 π}{65} c o s \frac{4 π}{65} c o s \frac{8 π}{65} c o s \frac{16 π}{65} c o s \frac{32 π}{65} = \frac{2. s i n \frac{2 π}{65}}{2 s i n \frac{π}{65}} . c o s \frac{2 π}{65} c o s \frac{4 π}{65} c o s \frac{8 π}{65} c o s \frac{16 π}{65} c o s \frac{32 π}{65} = \frac{2. s i n \frac{4 π}{65}}{2.4 s i n \frac{π}{65}} . c o s \frac{4 π}{65} c o s \frac{8 π}{65} c o s \frac{16 π}{65} c o s \frac{32 π}{65}$

$= \frac{2. s i n \frac{8 π}{65}}{2.8 s i n \frac{π}{65}} . c o s \frac{8 π}{65} c o s \frac{16 π}{65} c o s \frac{32 π}{65} = \frac{2. s i n \frac{16 π}{65}}{2.16 s i n \frac{π}{65}} c o s \frac{16 π}{65} . c o s \frac{32 π}{65}$

$= \frac{2. s i n \frac{32 π}{65}}{2.32 s i n \frac{π}{65}} . c o s \frac{32 π}{65} = \frac{s i n \frac{64 π}{65}}{64. s i n \frac{π}{65}} = . \frac{1}{64} . \frac{s i n (π - \frac{π}{65})}{s i n \frac{π}{65}} = \frac{1}{64} \frac{s i n \frac{π}{65}}{64 s i n \frac{π}{65}} = \frac{1}{64} = R H S$

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Similar questions

Q. Find :

cos \frac{π}{65} cos \frac{2 π}{65} cos \frac{4 π}{65} cos \frac{8 π}{65} cos \frac{16 π}{65} cos \frac{32 π}{65}

The value of $c o s \frac{π}{65} c o s \frac{2 π}{65} c o s \frac{4 π}{65} c o s \frac{8 π}{65} c o s \frac{16 π}{65} c o s \frac{32 π}{65}$ is

Q. Find the value of :

c o s \frac{π}{33} \cdot c o s \frac{2 π}{65} \cdot \frac{4 π}{65} \cdot c o s \frac{8 π}{65} \cdot c o s \frac{16 π}{65} \cdot c o s \frac{32 π}{65}

Q. The following number are given-

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R = \frac{(30^{65} - 29^{65})}{(30^{64} + 29^{64})}

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cos π65 cos 2π65 cos 4π65 cos 8π65 cos 16π65 cos 32π65=164

$c o s \frac{π}{65} c o s \frac{2 π}{65} c o s \frac{4 π}{65} c o s \frac{8 π}{65} c o s \frac{16 π}{65} c o s \frac{32 π}{65} = \frac{1}{64}$