Geometry with differential.

Hello teacher~

Curve a(t) = ( sin(3t).cos(t) , sin(3t).sin(t) , 0 )

I want to show that a(t) is a regular curve.
Namely, a'(t) =/= 0 for all t in R.

a'(t) = ( 3.cos(3t).cos(t) - sin(3t).sin(t) , 3.cos(3t).sin(t) +
sin(3t).cos(t) , 0 )

Can you show it without computer ?
Namely,
For each t in R,
[3.cos(3t).cos(t) - sin(3t).sin(t)] =/= 0
or [3.cos(3t).sin(t) + sin(3t).cos(t)] =/= 0.


For reference,
sin(3t) = 3.sin(t) - 4.{sin(t)}^3
cos(3t) = 4.{cos(t)}^3 - 3.cos(t)
.

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