Universal Formula of Trigonometric Function:

\[\sin{2\alpha}=\frac{2\tan\alpha}{1+\tan^2\alpha} \qquad \cos{2\alpha}=\frac{1-\tan^2\alpha}{1+\tan^2\alpha} \qquad \tan{2\alpha}=\frac{2\tan\alpha}{1-\tan^2\alpha}\]

Inclusion-Exclusion Principle:

\[\left\| \bigcup_{i=1}^n S_n \right\| = \sum_{k=1}^n (-1)^{k+1} \sum_{1 \le a_1 \lt a_2 \lt \cdots \lt a_k \le n} \left\| \bigcap_{i=1}^k S_{a_i} \right\|\]

Euler's Formula:

\[\mathrm{e}^{i\pi} + 1=0\]

Newton-Leibniz Formula:

\[\int_a^b f(x) \,\mathrm{d}x = F(b) - F(a)\]

Dirichlet Convolution:

\[(f*g)(n) = \sum_{k \mid n} f(k)g(\frac{n}{k}) = \sum_{k\ell=n} f(k)g(\ell)\]

Laplace Transform:

\[\hat{x}(s)=\int_0^\infty e^{-st} x(t) \,\mathrm{d}t\]