How did Gottlob Frege use Venus to show that people can think differently about the same object?

Gottlob Frege used the example of the Morning Star and the Evening Star to illustrate his distinction between sense (Sinn) and denotation (Bedeutung), which explains how people can think differently about the same object.12

While both names refer to the same physical planet—Venus—they possess different senses, or "modes of presentation". Because these names present the object in different ways, a person could know the meaning of both terms and observe the planet at different times without realizing they are identical.12

This distinction allows for informative statements of identity; for instance, discovering that the Morning Star is the Evening Star represents a genuine astronomical insight rather than a trivial tautology. By separating the referent (the object itself) from the sense (the way the object is presented to the mind), Frege demonstrated that thoughts are determined by these distinct modes of presentation, even when the underlying object remains the same.12