Cost vs. Costs when to use [closed]

Solution 1:

This is a question of whether you are talking about the notional or monetary price of secrets.

Cost noun
1 An amount that has to be paid or spent to buy or obtain something:
we are able to cover the cost of the event health-care costs
[mass noun]: the tunnel has been built at no cost to the state
1.1 The effort, loss, or sacrifice necessary to achieve or obtain something: the government succeeded in diverting resources away from consumption at considerable cost to its political popularity
- ODO

Your example is closest to the dictionary's example in definition 1.1, and that would be how the phrase secrets have their cost would normally be understood. Some may prefer the price of secrets. This idiomatically refers to intangibles such as loss of trust or the psychological burden of not sharing a secret with people affected by the secret. The phrase could also be uttered when the secret becomes known and problems result which could have been prevented by more transparency at an earlier time. Compare this with secrets take their toll and secrets extract a cost - both of these forms aren't idiomatic with tolls and costs.

Nevertheless, the plural form secrets have their costs is also grammatical. When referring to definition 1, it is a more pragmatic statement about costs that may be incurred, for example, for security measures. As Edwin Ashworth and Egox point out, it can also refer to definition 1.1 if several types of 'costs' are considered.