恒等式

1. 二项式展开

\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

常用特殊情形：

\[ (a+b)^2 = a^2 + 2ab + b^2 \]

\[ (a+b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3 \]

\[ (a+b)^4 = a^4 + 4a^3 b + 6a^2 b^2 + 4ab^3 + b^4 \]

反号版本：

\[ (a-b)^n = \sum_{k=0}^{n} (-1)^k \binom{n}{k} a^{n-k} b^k \]

\[ a^2 - b^2 = (a+b)(a-b) \]

\[ x^2 + y^2 = (x+y)^2 - 2xy \]

\[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \]

\[ x^3 - y^3 = (x-y)(x^2 + xy + y^2) \]

三项平方：

\[ a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab + bc + ca) \]

三项立方：

\[ (a+b+c)^3 = a^3+b^3+c^3 + 3(a+b+c)(ab+bc+ca) - 3abc \]

\[ a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1}) \]

奇次幂和公式：

\[ a^n + b^n = (a+b)(a^{n-1} - a^{n-2}b + \cdots - ab^{n-2} + b^{n-1}),\quad n \text{为奇数} \]

\[ x^3+y^3+z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2 - xy - yz - zx) \]

\[ (a+b)(b+c)(c+a) = ab(a+b) + bc(b+c) + ca(c+a) \]

\[ \sin^2 x + \cos^2 x = 1 \]

\[ \cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x -1 = 1 - 2\sin^2 x \]

\[ \sin 2x = 2\sin x \cos x \]

\[ \sin^2 x = \frac{1 - \cos 2x}{2} \]

\[ \cos^2 x = \frac{1 + \cos 2x}{2} \]

\[ \cos^3 x = \frac{3\cos x + \cos 3x}{4} \]

\[ \sin^3 x = \frac{3\sin x - \sin 3x}{4} \]

\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

\[ \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \]

\[ \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} \]

\[ 1 - x^n = (1-x)(1 + x + x^2 + \cdots + x^{n-1}) \]

\[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots,\quad |x|<1 \]