# Euler's formula for product of cosines

According to "Squaring the Circle" by Ernest Hobson, this formula $$\theta =\frac{\sin\theta}{\cos\frac\theta2\cdot \cos\frac\theta4\cdot \cos\frac\theta8\dots}$$ for $\theta<\pi$ is due to Euler. He gives no reference.

Does anyone knows the reference to the original Euler's work?

1. Likely the formula was obtained by applying $$\sin(2{\cdot}x)=2\cdot\sin x\cdot \cos x$$ recursively. (It seems to be the easiest way.) If you plug in $\theta=\tfrac\pi2$ then you get the Vieta's identity $$\frac\pi2= \frac2{\sqrt{2}}\cdot \frac2{\sqrt{2+\sqrt2}}\cdot \frac2{\sqrt{2+\sqrt{2+\sqrt2}}} \ldots$$ This formula seems to be the first known "explicit" expression for $\pi$.

2. So far it seems that Hobson (and others after Hobson, including Cajori) made a mistake by attributing this formula to Euler.

This formula is an immediate consequence of the product expansion of the sine: $$\frac{\sin x}{x}=\prod_{n=1}^\infty\left(1-\frac{x^2}{\pi^2n^2}\right),$$ and of the well-known fact that every integer is a product of a power of $2$ and an odd integer.
• I doubt that this is correct reference. The formula I am asking about is an immediate consequence of $$\sin (2{\cdot} x)=2\cdot\sin x\cdot\cos x.$$ One should be mad to get the formula the way you suggest. – Anton Petrunin Nov 30 '15 at 10:42