Cool! I've just contributed several examples. If anyone is interested in the sheer amount of identities that have been discovered, good books are (many of them gigantic references spanning thousands of pages). When bored, try proving some of those facts, examples build on top of each other. These are not the only examples, as there are many texts like these in other areas of mathematics and engineering, be it numerical analysis, optimization and variational analysis, statistics, abstract algebra, control theory, geometry and so on.

Table of Integrals, Series, and Products, Gradshteyn & Ryzhik.

Special Integrals of Gradshteyn and Ryzhik, Vols. I and II, Moll for some proofs of the above.

Handbook of Integral Equations, Polyanin & Manzhirov.

Scalar, Vector, and Matrix Mathematics, Bernstein.

Handbook of Number Theory I and II, Sandor, Crstici & Mitrinovic.

Wikipedia also has a plethora of pages with mathematical identities. Some of them:

https://en.wikipedia.org/wiki/Vector_calculus_identities

https://en.wikipedia.org/wiki/Vector_algebra_relations

https://en.wikipedia.org/wiki/Exterior_calculus_identities

https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spheric...

https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian...

https://en.wikipedia.org/wiki/List_of_set_identities_and_rel...

https://en.wikipedia.org/wiki/List_of_triangle_inequalities

https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...

... and its several lists of integrals (including trigonometric, exponential, rational).

https://en.wikipedia.org/wiki/Lists_of_integrals#Lists_of_in...

More advanced topics:

http://proximity-operator.net/