Gaussian binomial coefficients

Definition:The Gaussian binomial coefficient is given by

$${m\choose r}_q=\frac{(1-q^m)(1-q^{m-1})\dots(1-q^{m-r+1})}{(1-q)(1-q^2)\dots(1-q^r)}.$$

Exercise:Prove the following identities

$${m \choose r}_1={m\choose r},$$ (3)

$${m \choose r}_q=q^r{m-1 \choose r}_q+{m-1\choose r-1}_q = {m-1 \choose r}_q+q^{m-r}{m-1\choose r-1}_q,$$ (4)

$$\prod_{i=0}^{n-1}(1+q^it)=\sum_{i=0}^nq^{i(i-1)/2}{n\choose i}_qt^i,$$ (5)

$$\prod_{i=0}^{n-1}\frac 1{1-q^it}=\sum_{i=0}^\infty {n+i-1 \choose i}_qt^i.$$ (6)

The identity 2 shows that Gaussian binomial coefficients are generalizations of usual binomial coefficients. The identities 3 are called Pascal idenitites, the identity 4 is called Newton binomial formula. Use one of the identities to show that Gaussian binomial coefficient is a polynomial in $q$. $\;\Box$