Amazed At The Expression? Even I was too

Geometry Level 4

∏ k = 1 100 ( 1 + 2 cos ⁡ ( 2 π ⋅ 3 k 3 100 + 1 ) ) = ? \large\prod^{100}_{k=1} \left(1+2\cos\left(\frac{2\pi \cdot 3^k}{3^{100}+1}\right)\right)= ?


The answer is 1.

Md Zuhair by Md Zuhair , 20, India

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2 solutions

Aditya Kumar
Jul 7, 2017

For a hint use the fact that (2cos2A +1) = sin(3A)/sinA.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you derive the fact?

Md Zuhair - 3 years, 11 months ago

using componendo dividendo

Sidharth Nair - 3 years, 10 months ago
Hana Wehbi
Jul 7, 2017

At some point the cos ⁡ \cos function is gonna go to zero, so we end up with the product of ones which gives an answer of 1 1 .

The above product can be split into the sum of two products. Now, for the product of cosines, we know for some k k , we are going to have the value of c o s = 0 cos = 0 , then the whole product is 0 0 too because there is one term which is 0 0 . The remaining is product of 1 s 1s which is 1 1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you please justify your statement?

Md Zuhair - 3 years, 11 months ago

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