A more specific explanation is better than a general explanation in the scientific sense exactly because it is more easily falsifiable. Your sentence

If I know a complex differentiable function has roots on all multiples of pi, Saying the function is satisfied by Sin is more plausible then saying it's satisfied by some analytic function.

is completely wrong, as the set containing the sine function is most certainly contained in the set of all analytic functions, making it more plausible that "some analytic function has roots at all multiples of pi" than to say the same of sine, assuming we do not already know a great deal of information about sine.

However, If I do an experiment, and measure something which is implied by A&B, then I would think "A&B becomes more plausible then A", Because A is more vague then A&B.

Plain and simply no. If evidence E implies A and B, formally E -> A&B, then seperately E -> A and E -> B are true, increasing the probability of both seperately, making your conclusion invalid.