I spent over an hour on that question alone and I feel I got it correct by luck.

The question seemed to be intentionally vague and “Englishly” complicated for no good reason, which is a shame since Gavin is usually fair with the exams.

Multiple assumptions had to be made regarding the fuel on board and wind speeds to reach an answer.

Nothing from Bobs practice exams, the book or PPE covered the level of sophistication that question got into, I feel more on multi leg PNRs should be covered in the book. 

I can only wish the best of luck to whoever gets that same one…

Afraid that you have been led up the garden path during your training. Happens to lots of people because a LOT of the training people really don't know what they don't know and, so, errors and lack of knowledge beget more folks with lack of knowledge and so it keeps going on.

What is the exact drift angle that has to be in order for us to find ETAS, 5 degrees or 10 degrees?

First point - finding ETAS has absolutely nothing to do with the drift angle's having some particular value or not, as the case may be.   

Background - There are two main types of pilot navigation computer:  The first is

(a) the Dalton (mostly now clones), often called the E6B, which term dates back to the USAAC/USAAF stores designation for the device developed by Philip Dalton back in pre-WWII days.   The Dalton is the device with the sliding bit displaying drift splays.  From your post, it is pretty evident that you are using a Dalton clone.  The other common device is

(b) the CR (now mostly clones - which doesn't have any sliding bit) which dates back to a device developed by Ray Lahr (a United Airlines pilot) for UA in the early 50's which, subsequently, was purchased by Elrey Jeppesen (also, at the time, a UA employee).   The story has become a bit clouded with mix-and-match terminology being introduced in more recent years.   You will see CRs styled along the lines of "E6B Circular".   These are plain old CR clones and have absolutely nothing to do with the E6B, which is a plain old Dalton.  My guess is that the term was introduced for some misguided perceived marketing advantage to pick up a few more sales.   

Although it is difficult to find a definitive story, my take is that Jeppesen, with his sideline embryonic aviation company, wanted a device which was "different" to Dalton's, for marketing differentiation.   Lahr, being an ex-USN pilot, would have known of the WWII German DR2 device developed by Knemeyer before WWII and used that as the starting point for the CR design.  Lahr didn't coin the ETAS term - that was introduced by Jeppesen (or his team) after he purchased the rights to Lahr's device in the mid-50s.

Now, these two devices each will solve the wind triangle, which is the aim when you need to figure out things like what heading you need to fly from A to B and what groundspeed you might end up achieving.   However, while using the same basic navigation triangle, each does the solving deed in a quite different way.   

The Dalton draws the navigation triangle and solves it geometrically by measuring lengths and angles.   Dead simple and works really well.

The CR does the trick trigonometrically which is a mathematical technique but ends up with the same answer.

It follows that there is no necessary link between the two, ie, if you are using a Dalton, you dance the Dalton two-step while, if you are using the CR, you do the CR jive.   There is no mix and match.   You use the appropriate solution protocol for whichever gadget you might choose to use.  Please, forget all about trying to incorporate a Dalton bit into the CR or vice versa.  A bit like driving a Mercedes or a BMW.  Both are nice, both do much the same thing (get you from A to B) but there is a bit of a difference driving one or the other. 

IMPORTANT BIT TO KNOW - the Dalton approach DOES NOT USE, HAS NO USE FOR, AND HAS ABSOLUTELY NO INTEREST IN, ETAS while the CR MUST ALWAYS FIGURE OUT ETAS to calculate an accurate geometric length needed to figure out groundspeed.   The CR ALWAYS USES ETAS WHILE THE DALTON NEVER USES ETAS.   

So, where does the 10° and 5° nonsense come from ?   Dead easy to answer.

Back in the day, when Jeppesen was first playing with his newly purchased device, he noted that, as the drift angle varied, so did the difference between TAS and ETAS.   As the drift angle got smaller, the difference got smaller.   Further, he noted that, for practical use in the aircraft, once the drift angle got down to less than, say, 10° or thereabouts, it didn't really matter whether the pilot did the correct calculation or roughed it out by assuming that ETAS was around about the same value as TAS.   So, the Jeppesen and, subsequently, clone user manuals, all suggested that the pilot ignore the correct solution and just use TAS in lieu of ETAS if the drift angle were less than about 10°.

However, if you do this when you want an accurate answer, like in the exam, you will come unstuck.  More importantly, using this approximation is pretty stupid, as it takes you all of about 1-2 seconds longer to figure out ETAS and get the CORRECT answer.    CASA, realising that this was the case, suggested in its advice to ATPL candidates, that they should NOT use the Jeppesen approximation.   However, to keep the peace with everyone who believed the approximation nonsense, CASA suggested that, if you really wanted to use it, then only use it with drift angles less than 5°.   For these very small drift angles, the difference between ETAS and TAS is very close to zero so it doesn't matter for the exams.   However, it is still stupid because it is super easy and quick to use the standard solution with ETAS.   From a functional point of view, you will round ETAS off to TAS at these small drift angles because that makes practical sense.    So that's where the 10° and 5° came from.

How can I use it to find the ETAS ?

Dead easy, although it is easier on older styles of the Dalton as they had a squared grid on the slide.   All you have to do is draw from the end of the TAS vector a right-angled intercept to the ground speed vector and the length of the intercept on the groundspeed vector is ETAS.  No more, no less.   However, while you can do that, it is of absolutely no value to you other than if you are looking at a question relating to the CR where ETAS is important and, for whatever strange reason, you want to work out the ETAS value.

I hear some people here saying that in this type of computer, the ETAS has already been accounted for

All that says is that the folks saying such stuff don't understand how the two devices work - ie, they are by rote users - step 1, step 2, step 3 ....   The Dalton has absolutely NO interest in, or use for, ETAS and, most certainly, doesn't calculate it anywhere in the Dalton solution to the wind triangle.   ETAS is ONLY of relevance to the CR's trigonometric solution.   Period !

While you can figure out ETAS (for whatever strange reason) it is of NO practical value for you to do so, IF you are using a Dalton device.

if I have to find it through calculation, can I use the wind component grid that is printed on it?

First, you don't need to find ETAS for a Dalton solution.  If you were using the CR, the CR's solution presents the trigonometric stuff right there for you to read off the value of ETAS just to the left of the TAS index mark.

Second, if you really want to calculate separately to the CR's solution protocol, just use the CR as it is - the multiplication is right there on the outside scale for you to read off.   The equation is ETAS = TAS x cosine (drift angle).  You have set the TAS index on the inner scale (which is just the 10 scale mark on the log scale) against the TAS value on the outer scale.   Then, when you move left to the drift angle on the inner scale to read the ETAS value on the outer scale, all you are doing is running that equation I gave above.

Third, the squared wind component grid (on the CR) is there to provide some information relating to the wind velocity part of the wind triangle - nothing to do with ETAS.   I am not sure what grid you are talking about on the Dalton, as most Daltons, these days, don't have the square grid which, originally, was on the sliding scale.   That won't be of any use to you in calculating ETAS as you would need to have the grid able to be positioned over the triangle to do the right-angled intercept as described above.   Perhaps you can explain just which grid you are referring to and we can offer further comment ?

1. What is the exact drift angle that has to be in order for us to find ETAS, 5 degrees or 10 degrees?

2) I am using the E6B (ASA) model. How can I use it to find the ETAS? Because I hear some people here saying that in this type of computer, the ETAS has already been accounted for; if so, does that mean I just put in my wind and velocity and turn the wheel to my heading and then slide it up or down to my given TAS and under the center of the graticule I find my GS and don't need to go through all the processes of finding the crosswind and all calculations or something else? Please, I need your guidance; I'm a bit confused 

Also, if I have to find it through calculation, can I use the wind component grid that is printed on it?

Thanks for the above feedback... 

could you please give me an example - when " when you have to use radio navigation aids to navigate to an alternate.. 

I can only think, when both GNSS unit has been removed in VHOZY...

or I am missing something...

TIA..