As ilovevolatility pointed out, the main application of the COS method is to price options. The initially proposed method simply approximates the probability density function by a cosine expansion using the characteristic function of the log spot price. So, if you know the characteristic function (say for exponential Levy models and many stochastic volatility models), you can easily price options and obtain the Greeks. As you said, you can also extract the probability distribution of the log spot price.
The COS method was originally developed by Fang and Oosterlee (2008) as part of Fang's PhD thesis. This paper deals with the pricing of European-style options. Fang and Oosterlee (2009) show how the method can be adapted to price ealy-exercise features.
Zhang and Oosterlee (2013) and Zhang and Oosterlee (2014) discuss the pricing of discretely monitored geometric and arithmetic Asian options (with early exercise features). This algorithm is named ASCOS and is part of Zhang's PhD thesis. The prices of continuously monitored option are obtained by Richardson extrapolation.
Leitao et al. (2018) develop a model-free approach which does not require an analytically known characteristic function. Instead, the coefficients $A_n$ representing the distribution of the terminal stock price are estimated based on historical values. Thus, they name this approach data driven cosine (ddCOS) method.