I would like to estimate Angle of Attack using trigonometric relations i.e AoA = arctan (w/u) airspeed components (i.e u, v, w). When I am considering body 'x' velocity ('u' component), is it necessary to use the Indicated Airspeed? Instead I would like to use Calibrated Airspeed. Is it acceptable? Could anyone please clear my doubt?
You must use the same quantities for both speeds. Usually, this will be TAS.
As you see, you are going to use a ratio of speeds: w/u. This ratio must be dimensionless, and furthermore, the quantities must be "compatible". For such purposes, you can't meaningfully divide IAS by TAS (or CAS), or, say, altitude by flight level, even though they are both expressed in kt (or ft). They are just different physical quantities.
So, if you "would like" to use CAS for u, you must use CAS for w. How do you get it?
If you are trying to ascertain AoA that way on a real airplane in flight, you'll have difficulties measuring w (body normal airspeed). There is no sensor for it.1 And if you were to imagine some sort of a vertical Pitot tube, its IAS would have very different instrumental errors to the "normal" forward IAS, making these two IASes "incompatible", and you'd need to calibrate it extensively (and thus use CAS).
But the next catch is to check what CAS actually tries to express. Normally, the pilot (as well as the aerodynamicist) is interested in the "flight path" (total) airspeed rather than in the "body forward" airspeed. IAS measures the latter, but the standard CAS conversion may include correction for the AoA, amongst other things. If this is the case, you'd need to use arcsin instead of arctan (assuming you can get "CAS w").
In practice, one uses this formula to obtain AoA when solving equations of motion numerically. In this case, naturally, you have TAS everywhere as the primary quantity. You don't even need CAS nor IAS except for indication to the pilot, and perhaps as an input to autopilot or other equipment if you are simulating it.
1 Note the VSI measures vertical speed (in ground, not body frame), and it's not airspeed at all.