[QUE/ME-02009] Successive Rotations

Read the following theorem of  Rodrigues and Hamilton taken from Whittaker,  Sec2-3.

  The theorem of Rodrigues and Hamilton.Any two successive rotations about a fixed point can be compounded intoa single rotation by means of a theorem, which may be stated as follows:Successive rotations about three concurrent lines fixed in space, through twice the angles of the planes formed by them, restore a body to its original position. For let the lines be denoted by OP, OQ, OR. Draw. Op, Oq, Orperpendicular to the planes QOR, ROP, POQ respectively. Then if a body isrotated through two right angles about Oq, and afterwards through two rightangles about Or, the position of OP is on the whole unaffected, while Oq ismoved to the position occupied by its image in the line Or; the effect is therefore the same as that of a rotation round OP through twice the anglebetween the planes PR and PQ, which we may call the angle RPQ. It follows that successive rotations round OP, OQ, OR through twice the angles RPQ, PQR, QRP, respectively, are equivalent to successive rotations through two right angles about the lines Oq, Or, Or, Op, Op, Oq; but the latter rotations will clearly on the whole produce no displacement; which establishes the theorem.
Quoted from Whittaker

Now solve  the following problem. Following  two rotations are preformed in succession\\ (i) rotation by angle \(\alpha\) about axis \(\hat{n}\);\\ (ii) rotation by angle \(\beta\) about axis \(\hat{m}\).\\ Find the angle and axis of rotation that will produce the same result as the combined effect of above two rotations.