Step 1 — read the modulation index from the equation. Compare the given signal with the standard AM form:

\(s(t)=A_c\left[1+\mu\cos\omega_m t\right]\cos\omega_c t\)

Matching term by term with \(s(t)=20\left[1+0.9\cos 2\pi\times10^{4}t\right]\cos 2\pi\times10^{6}t\) gives carrier amplitude A_c = 20 V, modulating frequency f_m = 10 kHz, carrier frequency f_c = 1 MHz, and modulation index

\(\mu = 0.9\)

Step 2 — what power efficiency means. In conventional AM (DSB-FC) the carrier itself carries no information; only the two sidebands do. Efficiency is therefore the fraction of total transmitted power that lies in the sidebands:

\(\eta = \dfrac{P_{SB}}{P_t} = \dfrac{\mu^{2}}{2+\mu^{2}}\)

This comes from \(P_t = P_c\left(1+\dfrac{\mu^{2}}{2}\right)\) and \(P_{SB} = P_c\dfrac{\mu^{2}}{2}\).

Step 3 — substitute μ = 0.9.

\(\eta = \dfrac{(0.9)^{2}}{2+(0.9)^{2}} = \dfrac{0.81}{2.81}\)

\(\eta = 0.2882 = 28.82\%\)

Interpretation. Even at 90 % modulation, under 29 % of the transmitted power does useful work — the rest is spent on the carrier. Efficiency peaks at only 33.3 % for μ = 1 (full modulation), and driving μ > 1 causes over-modulation and envelope distortion rather than any gain. This poor efficiency is exactly why DSB-SC and SSB were developed: suppressing the carrier raises efficiency to 100 %, at the cost of needing a coherent detector instead of a simple diode envelope detector.

Useful check with actual powers (for a 1 Ω reference): \(P_c = A_c^{2}/2 = 200\ \text{W}\), \(P_{SB}=P_c\mu^{2}/2 = 81\ \text{W}\), \(P_t = 281\ \text{W}\), so \(\eta = 81/281 = 28.82\%\) ✓.

Hence, the power efficiency of the AM signal is 28.82 %.