I want a book for the proofs in mathematics for example the proof for (cos^2 +sin^2 = 1)
Hey eisenheim.
I don't think you will find a single book with every single proof in it.
Typically what you will find are books that have proofs and results for a particular field of mathematics - whether it be high school mathematics, analysis, topology, statistical inference and so on.
If you are looking for proofs regarding things like cos^2 + sin^2 = 1 then I would recommend looking at high school mathematics text books (upper level later years) or first/second year university textbooks.
You realize that such a book immediately goes out of date, anytime something new is discovered? I'm pretty sure there isn't a publisher who can keep up. I have a book, which only covers ONE subject (Algebra) that is nearly 1,000 pages long. A book such as you describe would undoubtedly fill up an entire library.
A statement like: $\cos^2\theta + \sin^2\theta = 1$ is actually a kind of short-hand, for this statement:
Suppose we have a triangle that has one right angle, and another angle we will call $\theta$. We call $\sin\theta$ the ratio of the length of the opposite side to the length of the hypotenuse (the hypotenuse is "the long side", and the "opposite side" is the non-hypotenuse side that isn't one of the two sides forming our angle $\theta$), and we call $\cos\theta$ the ratio of the of the length of the adjacent side to the length of the hypotenuse (the adjacent side is the one of the two sides forming the angle $\theta$ that isn't the hypotenuse).
For some reason the squares of these two numbers are written $\cos^2\theta,\sin^2\theta$ instead of $(\cos\theta)^2,(\sin\theta)^2$. Let's call the lengths of the opposite side, adjacent side, and hypotenuse: $O,A,H$. Then our original statement:
$\cos^2\theta + \sin^2\theta = 1$ becomes:
$\left(\dfrac{O}{H}\right)^2 + \left(\dfrac{A}{H}\right)^2 = 1$.
Multiplying through by $H^2$ gives us:
$O^2 + A^2 = H^2$.
In THIS form, this is just the Pythagorean Theorem. So, if we want to prove our original statement, we need to then prove the Pythagorean theorem, on which it rests.
For various proofs of the Pythagorean Theorem, you can go here: Pythagorean Theorem and its many proofs
Some of these proofs rely on other geometrical proofs, so that a "complete" proof of our original identity, would be even longer. And that's JUST ONE PROOF. Every proof begets another one:
For example: $\cos^2\theta + \sin^2\theta = 1$ leads to: $\sec^2\theta - \tan^2\theta = 1$.
As much as I hate to tell you this, one cannot learn "all of math" the way you can learn "all about tic-tac-toe". There's too much ground to cover. At some point, you'll have to focus on the parts that INTEREST you.