On close-to-scalar one-parameter cosine families

It is shown that if two cosine families with values in a normed algebra with unity, both indexed by t   running over all real numbers, of which one consists of the multiples of the unity of the algebra by numbers of the form cos⁡atcos⁡at for some real a  , differ in norm by less than 8/(33) uniformly in t  , then these families coincide. For a≠0a≠0, the constant 8/(33) is optimal and cannot be replaced by any larger number.