Using characteristic functions to show trigonometric equalty

Question

How would I show the following using characteristic functions.

$$\frac{\sin(t)}t=\frac{\sin(t/2)}{t/2}\cos(t/2)$$

and

$$\frac{\sin(t)}t=\prod_{k=1}^{\infty}\cos(\frac{t}{2^k})$$

Answer 1

You could try to recursively evaluate the equality,

\begin{eqnarray} \frac{\sin t}{t} &=& \color{blue}{\frac{\sin(t/2)}{(t/2)}}\cos(t/2) \\ &=& \color{blue}{\frac{\sin(t/4)}{(t/4)}\cos(t/4)}\cos(t/2) = \frac{\sin(t/2^2)}{t/2^2}\cos(t/2)\cos(t/2^2) = \frac{\sin(t/2^2)}{t/2^2}\prod_{k=1}^2\cos \left(\frac{t}{2^k}\right) \\ &\vdots& \\ &=& \frac{\sin (t/2^n)}{t/2^n}\prod_{k=1}^n\cos \left(\frac{t}{2^k}\right) \tag{1} \end{eqnarray}

Now take the limit $n \to \infty$, in this case $t/2^n\to 0$ and

$$ \lim_{n\to\infty} \frac{\sin (t/2^n)}{t/2^n} = \lim_{u\to 0}\frac{\sin u}{u} = 1 \tag{2} $$

Evaluating that in (1) you get

$$ \frac{\sin t}{t} = \prod_{k=1}^\infty\cos \left(\frac{t}{2^k}\right) $$