[UPSC IFoS 2025] Angular momentum, spin etc (Q247, 8M)

The raising (J_+) and lowering (J_-) operators are defined by J_+ = J_x + i J_y and J_- = J_x - i J_y. Show that-- (i) [J_z, J_\pm] = \pm \hbar J_\pm (ii) J_+ J_- = J^2 - J_z^2 + \hbar J_z