Prove the identity

Question

$$\cos \frac{x}{2} \cdot \cos \frac{x}{4} \cdot \cos \frac{x}{8} = \frac{\sin x}{ 8\sin \frac{x}{8}}$$

Conjecture a generalization of this result and prove its correctness by induction.

Ps: I have tried using identities, but I keep running on a loop. I wanted to use identities first to have an idea of what I have to do to generalize. Any ideas will be gladly appreciate it!

An equivalent question: http://math.stackexchange.com/q/8439/1242 — Hans Lundmark, Oct 08 '13 at 21:36

score 3 · Accepted Answer · answered Oct 08 '13 at 18:42

3

We have

$$\cos \frac{x}{2} \cdot \cos \frac{x}{4} \cdot \cos \frac{x}{8}\times 8 \sin \frac{x}{8}=4\cos \frac{x}{2} \cdot \cos \frac{x}{4} \cdot\sin\frac{x}{4}=2\cos \frac{x}{2} \cdot\sin \frac{x}{2} =\sin x$$

The generalization is $$\prod_{k=1}^n\cos \frac{x}{2^k} =\frac{\sin x}{2^n\sin \frac{x}{2^n} }$$

answered Oct 08 '13 at 18:42

It seems that you are busy these days as I but +1 makes you free. ;-) – Mikasa Oct 11 '13 at 17:22
@BabakS. What do you think of a person who teaches 24 hours a week and probably this requires much time to prepare the course of each session all this makes me very busy but Luckily I found in this site a lovely friend who makes me free :) – Oct 11 '13 at 19:31
The same story is for me, but I do 23 hours a week ;-) Great job Sami. :-) – Mikasa Oct 12 '13 at 12:56

score 2 · Answer 2 · answered Oct 08 '13 at 18:26

2

HINT:

$$\sin2A=2\sin A\cos A$$

Put $\displaystyle A=\frac x8,\frac x4,\frac x2$ in succession

answered Oct 08 '13 at 18:26

lab bhattacharjee

274,582

Prove the identity

2 Answers2