Some curious binomial coefficient identities I was playing around with some polynomials involving binomial coefficients, and inadvertently proved the following two identities:
(1) For all $p, q \in \mathbb{N}$ and $i \in \left[ 0, \min\{p, q\} \right]$:
$$
\begin{pmatrix}p \\ i\end{pmatrix} = \sum_{j=0}^i (-1)^j \begin{pmatrix}q \\ j\end{pmatrix} \begin{pmatrix}p + q - j \\ i - j\end{pmatrix} \text{.}
$$
(2) For all $q \in \mathbb{N}_{\ge 1}$ and $i \in [0, q]$:
$$
\frac{q - i}{q} = \sum_{j=0}^i (-1)^j \begin{pmatrix}i \\ j\end{pmatrix} \begin{pmatrix}2q - 1 - j \\ i - j\end{pmatrix} \begin{pmatrix}q - j \\ i - j\end{pmatrix}^{-1}  \text{.}
$$
Can either of these identities be proven in any trivial way (e.g., by reduction to known identities)?

For what it's worth in verifying that
$${p\choose r} = \sum_{j=0}^r (-1)^j 
{q\choose j} {p+q-j\choose r-j}$$
we may introduce
$${p+q-j\choose r-j} = {p+q-j\choose p+q-r} =
\frac{1}{2\pi i}
\int_{|z|=\epsilon} 
\frac{1}{z^{r-j+1}}
\frac{1}{(1-z)^{p+q+1-r}}
\; dz$$
which vanishes when $j\gt r$ so  we may extend $j$ to infinity and get
for the sum
$$\frac{1}{2\pi i}
\int_{|z|=\epsilon} 
\frac{1}{z^{r+1}}
\frac{1}{(1-z)^{p+q+1-r}}
\sum_{j\ge 0} (-1)^j {q\choose j} z^j
\; dz
\\ = \frac{1}{2\pi i}
\int_{|z|=\epsilon} 
\frac{1}{z^{r+1}}
\frac{1}{(1-z)^{p+q+1-r}}
(1-z)^q
\; dz
\\ = \frac{1}{2\pi i}
\int_{|z|=\epsilon} 
\frac{1}{z^{r+1}}
\frac{1}{(1-z)^{p+1-r}}
\; dz$$
This evaluates by inspection to
$${r+p-r\choose p-r} = {p\choose p-r} = {p\choose r}.$$
