Prove that - cos-5cos2 -5cos4 -5cos8 -5-1-16

L.H.S cos π/5 cos 2 π/5 cos 4 π/5 cos 8 π/5 ⇒ cos A cos 2A cos 2^2A cos 2^3 A, where A=π/5 ⇒ sin 2^4A/2^4 sin A=sin 16A/16 sin A ⇒ sin 16 π/5/16 sin π/5= ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Multiples of an Angle

Prove that : ...

Question

Prove that : $c o s \frac{π}{5} c o s \frac{2 π}{5} c o s \frac{4 π}{5} c o s \frac{8 π}{5} = - \frac{1}{16}$

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Solution

L.H.S $c o s \frac{π}{5} c o s \frac{2 π}{5} c o s \frac{4 π}{5} c o s \frac{8 π}{5}$

$\Rightarrow c o s A c o s 2 A c o s 2^{2} A c o s 2^{3} A, where A = \frac{π}{5} \Rightarrow \frac{s i n 2^{4} A}{2^{4} s i n A} = \frac{s i n 16 A}{16 s i n A} \Rightarrow \frac{s i n \frac{16 π}{5}}{16 s i n \frac{π}{5}} = \frac{s i n (3 π + \frac{π}{5})}{16 s i n \frac{π}{5}} = \frac{- s i n \frac{π}{5}}{16 s i n \frac{π}{5}} = - \frac{1}{16}$

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Similar questions

Q. Evaluate:

cos \frac{π}{5} cos \frac{2 π}{5} cos \frac{4 π}{5} cos \frac{8 π}{5} =

Q. Solve:

cos \frac{π}{5} cos \frac{2 π}{5} cos \frac{4 π}{5} cos \frac{8 π}{5}

Q. The value of

cos (π / 5) cos (2 π / 5) cos (4 π / 5) cos (8 π / 5)

Q. Find the value of the expression:

cos \frac{π}{5} cos \frac{2 π}{5} cos \frac{4 π}{5} cos \frac{8 π}{5}

cos \frac{π}{5} cos \frac{2 π}{5} = \frac{1}{4} a n d c o s \frac{π}{5} + cos \frac{3 π}{5} = \frac{1}{2}

without using standard value

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Prove that : cos π5 cos2π5 cos 4π5 cos 8π5=−116

L.H.S cos π5 cos2π5 cos 4π5 cos 8π5 ⇒ cos A cos 2A cos 22A cos 23 A, where A=π5⇒ sin 24A24 sin A=sin 16A16 sin A⇒ sin16π516 sin π5=sin(3π+π5)16 sin π5=−sinπ516 sin π5=−116

Prove that : $c o s \frac{π}{5} c o s \frac{2 π}{5} c o s \frac{4 π}{5} c o s \frac{8 π}{5} = - \frac{1}{16}$