= Solution
Differentiate the <mechanical energy> and use the <damped pendulum> equation:
$$
\boxed{\frac{dE}{dt}=\dot\theta(\ddot\theta+\sin\theta)=-c\dot\theta^2\leq0.}
$$
The <energy dissipation of a damped pendulum> makes $E$ nonincreasing, rather than strictly decreasing at every instant: its derivative vanishes at turning points. It is constant along an equilibrium trajectory. Along every nonstationary trajectory it strictly decreases over any nonzero time interval, because a vanishing integral of $c\dot\theta^2$ over an interval would force an equilibrium there and hence everywhere by uniqueness.
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