Prove the product of terms

Question

Prove that

$$\left(1+x^{-1}\right)\cdot\left(1+x^{-2}\right)\cdot\left(1+x^{-4}\right)\dots\dots\left(1+x^{-2^n}\right)=\frac{x}{x-1}\cdot\left(1-\frac{1}{x^{2^{(n+1)}}}\right)$$

I thought a lot about this, but didn't get any breakthroughs. Please help me here.

Answer 1

Hint: Multiply and divide the LHS by $(x-1)$. And use the identity $(a-b)(a+b)=(a^{2}-b^{2})$.

Answer 2

$$F_n=(1+x^{-1})(1+x^{-2})(1+x^{-4}).......(1-x^{2^n})$$ Multiply by $(1-x^{-1})$ on both sides, then $$F_n \frac{1-x}{x}=(1-x^{-1}) (1+x^{-2}) (1+x^{4}).....(1-x^{-2^n})$$ Using $(a-b)(a+b)=(a^2-b^2)$ repetedly we get $$F_n=\frac{x}{1-x} (1-x^{-2^{n+1}})$$