大一/高数/等价变形

$$x^2-a=0\Rightarrow x={\color{red}\pm a(a\ge0)}\\ \sqrt{1-\cos^2x}=|\sin x|\\ \sqrt{1-\cos x}\sim\frac {\sqrt{2}}{2} {\color{red}{|x|}}\\ (\sqrt{a})^2=a,(\sqrt{a^2})=|a|\\ \sqrt{\frac AB},讨论A,B正负\\ f(x)>x\Rightarrow\begin{cases}\frac{f(x)}x>1,x>0\\ \frac{f(x)}x<1,x<0\\ f(0)>0,x=0 \end{cases}$$ $$(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a \\ a^{2}+b^{2}+c^{2}+a b+b c+c a=\frac{1}{2}\left[(a+b)^{2}+(b+c)^{2}+(c+a)^{2}\right] \\ a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right) \\ x^{n}-y^{n}=(x-y)\left(x^{n-1}+x^{n-2} y+\ldots+x y^{n-2}+y^{n-1}\right) \\ x^{2 n+1}+y^{2 n+1}=(x+y)\left(x^{2 n}-x^{2 n-1} y+x^{2 n-2} y^{2}-\ldots-x y^{2 n-1}+y^{2 n}\right) \\=(x+y) \cdot \sum_{i=0}^{2 n}\left[x^{2 n-i}(-y)^{i}\right] \\ x y+n x+m y+m n=(x+m)(y+n) \\ x y z+x y+y z+z x+(x+y+z)+1=(x+1)(y+1)(z+1) \\ x^{2} y+y^{2} z+z^{2} x+z^{2} y+y^{2} x+x^{2} z+2 x y z=(x+y)(y+z)(z+x) \\ x^{2} y+y^{2} z+z^{2} x+z^{2} y+y^{2} x+x^{2} z+3 x y z=(x+y+z)(x y+y z+z x) \\ x^{3}+y^{3}+z^{3}-3 x y z=(x+y+z)\left(x^{2}+y^{2}+z^{2}-x y-y z-z x\right) \\ =(x+y+z) \cdot \frac{1}{2}\left[(x-y)^{2}+(y-z)^{2}+(z-x)^{2}\right] \\$$ $$e=\sum_{i=0}^\infty \frac1{i!}\\ x=e^{\ln x}=\ln e^x$$ $$x+\sqrt{x^2+1}=C\\ \frac 1C=\sqrt{x^2+1}-x$$ $$a^2+b^2=c^2\\ 3,4,5\quad5,12,13\\ 8,15,17\quad7,24,25\\9,40,41$$