I have several books of proofs such as:
$\color{blue}{\text{Book of Proof}}\text{ by }\color{green}{\text{Richard Hammack}}$
$\color{blue}{\text{Proofs: A Long-Form Mathematics Textbook}}\text{ by }\color{green}{\text{Jay Cummings}}$
and some other books in proofs. These are really excellent books, and there are many proof books that are good.
However I am looking for a book that contains many (basic) or (common) or (concrete, possibly hard)
$$\color{red}{\text{the following are just examples}}$$
There are infinitely many primes.
The irrationality of $\sqrt{2}, e, \pi, \dots$.
Derivation of quadratic formula, cubic formula, quartic formula.
There is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients (also known as Abel–Ruffini theorem).
The area of a circle $A=\pi r^2$, the volume of a sphere $V=\frac{4}{3}\pi r^3, \dots$.
The Pythagoras' theorem, $a^2+b^2=c^2$ where $a$ and $b$ are the legs, and $c$ is the hypotenuse of a triangle.
$\sin(\alpha \pm \beta)=\sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$, $\cos(\alpha \pm \beta) = \dots, \tan(\alpha \pm \beta) = \dots$.
Wilson theorem, any prime $p$ divides $(p-1)!+1$.
Fermat's little theorem.
Sine and cosine rules.
Intermediate value theorem, mean value theorem, Rolle's theorem, ...
The sum of the first $n$ natural numbers, the sum of the squares of the first $n$ natural numbers, the sum of the cubes of the first $n$ natural numbers.
The sum of $\text{A.P, G.P}$.
The derivation of Taylor series formula.
The integral of (even function or odd function) from $-a$ to $a$.
L'Hospital's rule.
The examples mentioned above are not all easy, for example Abel–Ruffini theorem, or the quartic formula, they are difficult. But I believe the are basic (basic in the sense of "knowing" the proofs/derivations).
So what I am looking for is a book that contain the proofs of formulae in different branches of mathematics, such as arithmetic, geometry, trigonometry, algebra, calculus, number theory, ...
I am afraid such book does not exist. But if exist, I highly recommend myself and everyone who is interested in maths to get this book.
Your help would be appreciated. THANKS!
