I will make an attempt to explain in my way.
Static friction is self adjusting in nature. This means whenever a force tries to move a body on a roguh surface, friction starts resisting this force as far as it can till the applied force overcomes the maximum static friction possible. Which in other words also means that during the time the applied force increases from zero to the value at which the body just starts moving, coefficient of static friction also increases from zero to the maximum possible value.
Now, with this concept in mind, just re look at your question. For a body to be in equilibrium on an inlcined plane of 45°, what is the only condition ?
The answer is: The force which tries to move the body down the plane should be at all times less than the maximum value of static friction (i.e.the maximum force friction can resist). This mathematically looks like:
mgsinø < μN where μ= Maximum value of static friction.
Now as there is no motion perpendicular to the inclined plane, you can always use
N= mgcosø and substitute in the above equation to get:
mgsinø < μmgcosø i.e. tanø < μ or μ > tanø. Now as ø=45°, you arrive at your answer:
**μ > 1**
To summarize: your starting equation itself is with greater than/less than sign and not equality sign which was your only confusion. The equality sign equation here represents only the case where the applied force is equal to the maximum static friction possible or motion is impending or the body is just about to move which is something you dont want to analyse to answer your question.