Final answer:
The assertion that the current in the inductor varies sinusoidally when connected to a charged capacitor is correct because the energy oscillates between the capacitor and inductor, resulting in a sinusoidal current. The reason is correct as well because the total energy is conserved in the absence of resistance.
Step-by-step explanation:
When a charged capacitor is connected to an inductor, the current in the inductor does indeed vary sinusoidally. This happens because the energy initially stored in the capacitor's electric field is transferred to the inductor's magnetic field and vice versa. This results in an oscillating current and voltage, which gives us a sinusoidal waveform. As for the total energy of the circuit, it remains constant if we ignore resistance, due to the conservation of energy. However, in a real circuit, some energy would be lost due to resistance, even if it is small.
Given this explanation, the correct answer is A: Both the assertion that the current in the inductor varies sinusoidally and the reason that the total energy of the circuit remains constant are correct, and the reason is the correct explanation of the assertion.