Challenge Level

Consider these three algebraic identities

$$(1-x)(1+x+x^2+x^3)\equiv (1-x^4)$$

and

$$(1-x)\big((1+x)(1+x^2)(1+x^4) \big)\equiv (1-x^8)$$

and

$$\cos\left(\frac{x}{2}\right)\cos\left(\frac{x}{4}\right)\cos\left(\frac{x}{8}\right)\cos\left(\frac{x}{16}\right)\equiv\frac{\sin(x)}{16\sin\left(\frac{x}{16}\right)}$$

Prove that they are true for any real number $x$ in the first two cases and any real numbers except multiples of $16\pi$ in the third case.

Use the ideas in your proofs to write down general forms

$$(1-x)(1+x+x^2+x^3+\dots+x^n)\equiv \quad ???$$

and

$$(1-x)\big((1+x)(1+x^2)(1+x^4)\dots(1+x^{2^n}) \big)\equiv \quad ???$$

and

$$\cos\left(\frac{x}{2}\right)\cos\left(\frac{x}{4}\right)\cos\left(\frac{x}{8}\right)\cos\left(\frac{x}{16}\right)\dots \cos\left(\frac{x}{2^n}\right)=\quad ???$$

In each case it is possible to write down 'infinite $n$' limiting identities. What are these, and for which values of $x$ are they valid?