Step 1: Analyze the equation for set P.

The set P is defined by the equation: $\sin \theta - \cos \theta = \sqrt 2 \cos \theta$

We can rearrange this equation to solve for $\tan \theta$.

$\sin \theta = \sqrt 2 \cos \theta + \cos \theta$ $\sin \theta = (\sqrt 2 + 1) \cos \theta$

To express this in terms of $\tan \theta$, we need to divide by $\cos \theta$. We must first ensure that \cos \theta eq 0. If $\cos \theta = 0$, the original equation becomes $\sin \theta - 0 = \sqrt 2 \times 0$, which implies $\sin \theta = 0$. However, $\sin \theta$ and $\cos \theta$ cannot both be zero for the same angle $\theta$, since $\sin^2 \theta + \cos^2 \theta = 1$. Therefore, \cos \theta eq 0.

Now, we can divide by $\cos \theta$: $\frac{\sin \theta}{\cos \theta} = \sqrt 2 + 1$ $\tan \theta = \sqrt 2 + 1$

So, set P is the set of all angles $\theta$ for which $\tan \theta = \sqrt 2 + 1$.

Step 2: Analyze the equation for set Q.

The set Q is defined by the equation: $\sin \theta + \cos \theta = \sqrt 2 \sin \theta$

We rearrange this equation as well.

$\cos \theta = \sqrt 2 \sin \theta - \sin \theta$ $\cos \theta = (\sqrt 2 - 1) \sin \theta$

To express this in terms of $\tan \theta$ or $\cot \theta$, we need to divide by $\sin \theta$. We must first ensure \sin \theta eq 0. If $\sin \theta = 0$, the original equation becomes $0 + \cos \theta = \sqrt 2 \times 0$, which implies $\cos \theta = 0$. As established before, this is not possible. Therefore, \sin \theta eq 0.

Now, we can divide by $\sin \theta$: $\frac{\cos \theta}{\sin \theta} = \sqrt 2 - 1$ $\cot \theta = \sqrt 2 - 1$

To compare this with the condition for set P, we find $\tan \theta$: $\tan \theta = \frac{1}{\cot \theta} = \frac{1}{\sqrt 2 - 1}$

To simplify this expression, we rationalize the denominator: $\tan \theta = \frac{1}{\sqrt 2 - 1} \times \frac{\sqrt 2 + 1}{\sqrt 2 + 1} = \frac{\sqrt 2 + 1}{(\sqrt 2)^2 - 1^2} = \frac{\sqrt 2 + 1}{2 - 1} = \sqrt 2 + 1$

So, set Q is the set of all angles $\theta$ for which $\tan \theta = \sqrt 2 + 1$.

Step 3: Compare sets P and Q.

From Step 1, we found that any element $\theta$ in P satisfies $\tan \theta = \sqrt 2 + 1$. From Step 2, we found that any element $\theta$ in Q satisfies $\tan \theta = \sqrt 2 + 1$.

Since both sets are defined by the exact same condition, they must contain the exact same elements. Therefore, the set P is equal to the set Q. $P = Q$

Step 4: Evaluate the given options.

A: $P \subset Q$ and $Q - P \neq \emptyset$. This means P is a proper subset of Q, which is false because P=Q implies $Q-P = \emptyset$. B: $Q \not\subset P$. This is false because P=Q implies $Q \subseteq P$. C: $P \not\subset Q$. This is false because P=Q implies $P \subseteq Q$. D: P = Q. This is true, as we have shown.

Thus, the correct option is D.