Pattern Discovery & Generalization
Try inputs like:
sin(2x), sin(3x), sin(4x)
TrigExpand[sin(2x)]
TrigExpand[sin(3x)]
TrigExpand[sin(4x)]
What pattern do you see in how sine is expanded?
Define a sequence:
Sequence[TrigExpand[sin(n x)], n, 1, 5]
What if you tried cos(2x), cos(3x)?