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OP's eq. (1) is true for any invertible coordinate transformation -- they don't need to be canonical coordinates. The trick is to keep track of what is kept constant during the partial differentations. In 2D eq. (1) reads:

$$\begin{align} {\rm LHS} ~=~&\left(\frac{\partial Q}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_q \left(\frac{\partial Q}{\partial p}\right)_p\cr ~=~& \left[\left(\frac{\partial q}{\partial q}\right)_p \left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q \cr &- \left(\frac{\partial P}{\partial q}\right)_p \left[\left(\frac{\partial q}{\partial p}\right)_q \left(\frac{\partial Q}{\partial q}\right)_P +\left(\frac{\partial P}{\partial p}\right)_q \left(\frac{\partial Q}{\partial P}\right)_q \right]\cr ~=~& \left[\left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q \left(\frac{\partial Q}{\partial P}\right)_q \cr ~=~& \left(\frac{\partial Q}{\partial q}\right)_P\left(\frac{\partial P}{\partial p}\right)_q ~=~\left(\frac{\partial Q}{\partial q}\right)_P / \left(\frac{\partial p}{\partial P}\right)_q ~=~{\rm RHS},\end{align}$$ where we used the multi-variable chain rule twice.

OP's eq. (1) is true for any invertible coordinate transformation -- they don't need to be canonical coordinates. The trick is to keep track of what is kept constant during the partial differentations. In 2D eq. (1) reads:

$$\begin{align} {\rm LHS} ~=~&\left(\frac{\partial Q}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial p}\right)_q\cr ~=~& \left[\left(\frac{\partial q}{\partial q}\right)_p \left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q \cr &- \left(\frac{\partial P}{\partial q}\right)_p \left[\left(\frac{\partial q}{\partial p}\right)_q \left(\frac{\partial Q}{\partial q}\right)_P +\left(\frac{\partial P}{\partial p}\right)_q \left(\frac{\partial Q}{\partial P}\right)_q \right]\cr ~=~& \left[\left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q \left(\frac{\partial Q}{\partial P}\right)_q \cr ~=~& \left(\frac{\partial Q}{\partial q}\right)_P\left(\frac{\partial P}{\partial p}\right)_q ~=~\left(\frac{\partial Q}{\partial q}\right)_P / \left(\frac{\partial p}{\partial P}\right)_q ~=~{\rm RHS},\end{align}$$ where we used the multi-variable chain rule twice.

OP's eq. (1) is true for any invertible coordinate transformation -- they don't need to be canonical coordinates. The trick is to keep track of what is kept constant during the partial differentations. In 2D eq. (1) reads:

$$\begin{align} {\rm LHS}~=~&\left(\frac{\partial Q}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_q \left(\frac{\partial Q}{\partial p}\right)_p\cr ~=~& \left[\left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q \left(\frac{\partial Q}{\partial P}\right)_q \cr ~=~& \left(\frac{\partial Q}{\partial q}\right)_P\left(\frac{\partial P}{\partial p}\right)_q ~=~\left(\frac{\partial Q}{\partial q}\right)_P / \left(\frac{\partial p}{\partial P}\right)_q ~=~{\rm RHS},\end{align}$$ where we used the multi-variable chain rule twice.

OP's eq. (1) is true for any invertible coordinate transformation -- they don't need to be canonical coordinates. The trick is to keep track of what is kept constant during the partial differentations. In 2D eq. (1) reads:

$$\begin{align} {\rm LHS} ~=~&\left(\frac{\partial Q}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_q \left(\frac{\partial Q}{\partial p}\right)_p\cr ~=~& \left[\left(\frac{\partial q}{\partial q}\right)_p \left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q \cr &- \left(\frac{\partial P}{\partial q}\right)_p \left[\left(\frac{\partial q}{\partial p}\right)_q \left(\frac{\partial Q}{\partial q}\right)_P +\left(\frac{\partial P}{\partial p}\right)_q \left(\frac{\partial Q}{\partial P}\right)_q \right]\cr ~=~& \left[\left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q \left(\frac{\partial Q}{\partial P}\right)_q \cr ~=~& \left(\frac{\partial Q}{\partial q}\right)_P\left(\frac{\partial P}{\partial p}\right)_q ~=~\left(\frac{\partial Q}{\partial q}\right)_P / \left(\frac{\partial p}{\partial P}\right)_q ~=~{\rm RHS},\end{align}$$ where we used the multi-variable chain rule twice.

OP's eq. (1) is true for any invertible coordinate transformation -- they don't need to be canonical coordinates. The trick is to keep track of what is kept constant during the partial differentations. In 2D eq. (1) reads:

$$\begin{align} {\rm LHS}~=~&\left(\frac{\partial Q}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_q \left(\frac{\partial Q}{\partial p}\right)_p\cr ~=~& \left[\left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q \cr ~=~& \left(\frac{\partial Q}{\partial q}\right)_P\left(\frac{\partial P}{\partial p}\right)_q~=~\left(\frac{\partial Q}{\partial q}\right)_P / \left(\frac{\partial p}{\partial P}\right)_q~=~{\rm RHS}.\end{align}$$

OP's eq. (1) is true for any invertible coordinate transformation -- they don't need to be canonical coordinates. The trick is to keep track of what is kept constant during the partial differentations. In 2D eq. (1) reads:

$$\begin{align} {\rm LHS}~=~&\left(\frac{\partial Q}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_q \left(\frac{\partial Q}{\partial p}\right)_p\cr ~=~& \left[\left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q \left(\frac{\partial Q}{\partial P}\right)_q \cr ~=~& \left(\frac{\partial Q}{\partial q}\right)_P\left(\frac{\partial P}{\partial p}\right)_q ~=~\left(\frac{\partial Q}{\partial q}\right)_P / \left(\frac{\partial p}{\partial P}\right)_q ~=~{\rm RHS},\end{align}$$ where we used the multi-variable chain rule twice.