Angular displacement, minimum diameter of crater, and telescopes A telescope gives a clear view of a distant object when the angular displacement between the edges is at least $9.7 × 10^{-6} rad$. 
The Moon is approximately $3.8 × 10^5$ km from the Earth. Estimate the minimum diameter of a circular crater on the Moon's surface that can be seen using the telescope. 
The first way I did it is by using $tan (Q/2)$ = radius / distance from the Earth to the Moon.
The second way is by using $s = rQ$, where $s$ is the arc length ( which can be approximated to be the diameter since '$Q$' is small ) and $r$ is approximated to be the distance from the Earth to the Moon. 
They both give the same answer, but I think the method using trignometry is better since we assume less. The 'correct method', however, is the second one according to the answer. If the first were acceptable, I reckon it would've been mentioned. 
Is there something wrong with it? I thought that it was better since it worked for large angles too. 
Also, a far more difficult question I have is this : 
Suppose that we have 2 bodies $A$ and $B$  joined together into a 'circular crate'. Now, the angular displacement between the edges of the entire body is $10\mu rad$, which we assume to be the minimum angular displacement required for a clear view. Thus, we have a clear view of the entire body, which of course includes body A. 
Remove body B, and then note that the angular displacement between the edges of body A is less than $10\mu rad$. Thus we will not have a clear view of body A. 
But this is a contradiction, since we were earlier having a clear view of the entire body. How can removing body B affect the visibility of body A?
Why am I getting this absurd result?
https://i.stack.imgur.com/HwHwx.png
Edit:
Thanks to Sammy Gerbil, some parts of my question have been clarified. I, however, now have another question. 
In this diagram: https://i.stack.imgur.com/iI2nL.png
We cannot distinguish between the points $(A,A_0)$ or $(A,A_1)$ or $(A,A_2)$ and basically any 2 points which do not lie on the diameter. Then how is it that we can see the shape clearly? 
 A: You are wrongly interpreting the criterion. Objects do not suddenly become "clear" when the angle they subtend increases above $10\mu rad$. The criterion is only a rough indication of when details are too blurred to distinguish. There is a gradual blurring of the image when looking at smaller scales.
What the criterion means is that if the edges of A and B are greater than $10\mu rad$ apart then we can probably distinguish that they are two separate overlapping circles, whereas if the edges are less than $10\mu rad$ apart we cannot tell if we are seeing only one circle or 2 overlapping circles. In the latter case if you now remove A or B the image does not become clearer, and is only marginally more fuzzy : we still cannot tell if we are looking at 1 circle or 2 overlapping.

In the diagram above, the grid squares are $10\mu rad$ x $10\mu rad$. We cannot distinguish the difference between the large red and blue rectangles in the telescope, because the edges differ by less than the limit of $10\mu rad$. If they overlap we could not tell if we are seeing a single rectangle or two overlapping rectangles. The blurred images would look too similar for us to tell them apart.
Neither would we be able to tell if the red and blue circles were distinct circles or part of the same larger shape. Neither could we decide if the small red square is a different shape from the small red circle : the difference is less than the limit of $10\mu rad$ so we cannot distinguish the details. All that we could say here is that the small red circle and square are 2 separate objects.

In the illustration above, in the image on the left the resolution is about $0.3\mu m$. We can tell that we are not seeing a single circle, but we cannot tell if we are seeing a single object (eg a rhombus) or a composite figure (a group of spheres). If the resolution limit were say $1\mu m$ the image would be so blurred that we would not be able to tell if we were looking at a rhombus, square or circle. The image on the right is the same object viewed at a resolution of less than $0.1\mu m$. Now that we can see finer details we can tell that we are looking at a group of 6 spheres, although we cannot tell if they are joined together in some way - for that we would need a resolution of less than perhaps $0.03\mu m$.
A: 
Is there something wrong with it? I thought that it was better since it worked for large angles too.

No, there's nothing wrong with the formula using the tangent function. In fact, the other formula only works for small angles; it is an approximation of the tangent formula. That's because tangent can be expanded as a polynomial (see Maclaurin Series expansion):
$$\tan\theta = \theta+\frac{1}{3}\theta^3+\frac{2}{15}\theta^5 + \ldots $$
where $\theta$ is in radians. If $\theta$ is small ($<<1$), then the $\theta^3$ and higher power terms are small enough to ignore for most purposes. That means
$$\tan\theta \simeq \theta.$$
Applying that to your tangent formula we get
$$\tan\frac{Q}{2}=\frac{s}{2r}\simeq \frac{Q}{2}$$
which is, by the small angle approximation, the same as your second formula.  The trig method is correct.
