For the unknowns you dont know T, beta or R. So taking moments about C would be an obvious choice to eliminate both T and beta from the balance equation. If you resolved horizontally, vertically, along the plane, perpendicular to the rope, ... you wouldn't be able to eliminate both at the same time and would need more than one equation to solve simultaneously. Quesiton spotting a bit, you might note that they ask for beta in part b) so its a bit of a hint that its probably not necessary in a). If necessary, roughly write down each force balance equation at the start, and think about which one(s) simply the problem by eliminating unkown terms at the start.
For b) theres not an easy way (see below) to get beta without also considering T, but a bit of foresight would be that resolving vertically and horizontally (for instance) would give Tsin(beta-alpha) = ... and Tcos(beta-alpha) = ... as you know all the rest and then simply divide to get tan and eliminate T. You could do a variety of approaches though (along plane and perpendicular) or another moment equation and one projection as the ms says.
Edit - another way of thinking about this (less algebra/simultaneous equations) is to resolve the forces horizontally and vertically (without T) then reason that to balance, T must be the hypotenuse of the corresponding force triangle with muR being the horizontal leg and mg-R the vertical leg. So tan gives the angle with the horizontal and add alpha to get beta. Similarly if you wanted to find T without beta, resolve horizontally and vertically then use pythagoras. Its equivalent to the algebra using sin^2+cos^2=1 to combine the equations / eliminate beta
Note that for b) Id be tempted to resolve perpendicular to the string as that would give a single equation which would give beta, though there would be a bit more work setting up the angles/equation in the first place.
Guess the key thing is for forces youre not interested in, try resolving perpendicular to them or taking moments about the point at which theyre applied, rather than setting up "lots" of simultaneous equations and eliminating them. Be clear about which unknowns you want to find and which ones are not necessary.
For a few questions, why not work though a problem (rather than looking at the ms too soon) two or three different ways and thinking about the pros/cons of the different approaches. That way, its easier to appreciate the decisions you make at the start where youre trying to think a couple of steps ahead about which approach(es) are sensible.