SSB and Hilbert transform
You need to show this is the LOWER sideband.
Compare this, with the full math description of an AM signal.
Examine the AM signal, extract the carrier, the upper sideband, and the lower sideband.
Think of the two sidebands as Phasors, rotating, with Inphase and with Quadrature energy components.
Also the phasors may or may not be phase-aligned.
DRAW A POLAR PLOT of each phasor, and add arrows onto the phasor tip to define the direction of rotation; also become clean when the angles are identical and when angles do not agree.
Then decide if your trigonometric result is what you desire.
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Decades ago I taught a Transmitter Design newbie to implement GSM (GMSK) cellphone modulation. Turns out the GMSK (Gaussian minimum shift keying) is a precision single_sideband system, to avoid wasting energy in a carrier. Energy is much better used in reducing Bit Error Rate by improving the data_eye.
The SSB generation method used 0/90 oscillator and 0/90 modulation (4 signals) combined in 2 multipliers and then linearly summed.
Once we cleaned up the onchip capacitive parasitics, the Design produced 50dB carrier and 40+ dB unwanted_sideband suppression. Life was good.
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the only difference between AM and PM is the phasors underlying the two sidebands.

AM sidebands are identical in energy (so the trigonometry tells us), but the phasors in the two sidebands rotate in OPPOSITE directions, meeting only at top (90 degrees on a polar plot) and bottom (270 degrees). This rotation in opposite direction results in NO PHASE DEVIATION but plenty of amplitude variation.

PM sidebands are identical in energy, but the 2 sidebands are 180 degrees out of phase and rotate in the SAME direction. Result (of being 180 degrees out of phase) is that any inphase energy gets cancelled (thus no amplitude variation), but the orthogonal energy gets to add and become a useful phase variation.


So -------------------- how does ONE cause AM_to_PM conversion? Notice ANY low_pass_filtering (which ANY circuit will provide) results in some residual PM being generated, because the two sidebands no longer have IDENTICAL power. And the slight delay in any circuit, and the slightly different delay between USB and LSB, causes a PhaseShift between our two phasors, so the Quadrature energies no longer exactly cancel.
Thus for highest quality modulation, you need wideband circuits that construct the components of the trigonometric behaviors.
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Let us again view the generation of signals, at least single_tone signals, as an exercise in Phasors, where the Inphase and the Quadrature energies are crucial, and the phase tracking (or precise phase_anti_tracking --- exactly opposite ) is crucial.