How to drive a load that draws 1 A with a sinusoidal voltage?
You can use an op-amp and power transistors in a push-pull, emitter-follower configuration to get what you want: -



There is no need to bias the transistors (to avoid cross-over distortion) because, at such low operating frequencies, the overall negative feedback will rapidly push each transistor through that region.

And, if you know that the maximum |peak| voltage on the load is (say) 2 volts, the +/-12V rails to the transistors can be dropped to (say) +/- 3.3 volts or maybe even a shade lower. This will significantly reduce power dissipation in the output transistors.