Simplifying expressions. How do you simplify the following expression?
$$Q=(1-\tan^2(x)) \left(1-\tan^2 \left(\frac{x}{2}\right)\right)\cdots \left(1-\tan^2\left(\frac{x}{2^n}\right)\right)$$
I've tried the the following :
$\tan x = \frac{\sin x}{\cos x}$
$$\cos x = \frac{\sin 2x}{2\sin x}$$
$$\tan x = \frac{2\sin^2x}{\sin2x}$$
$$Q= \left( 1- \left(\frac{2\sin^2x}{\sin2x}\right)^2\right) \left(1-\left(\frac{2\sin^2\frac{x}{2}}{\sin x}\right)^2\right) \cdots \left(1-\left(\frac{2\sin^2\frac{x}{2^{n+1}}}{\sin(\frac{x}{2^n})}\right)^2\right)$$
But i have no idea how to continue.
Help appreciated!

Hint:
$$
1-\tan^2 t=\frac{\cos^2t-\sin^2t}{\cos^2t}=\frac{\cos(2t)}{\cos^2t}
$$
and for arguments of the form $\frac{x}{2^n}$, many things will vanish in the product.
