% Required to support mathematical unicode
\usepackage[warnunknown, fasterrors, mathletters]{ucs}
\usepackage[utf8x]{inputenc}

% Always typeset math in display style
\everymath{\displaystyle}

% Use a larger font size
\usepackage[fontsize=14pt]{scrextend}

% Standard mathematical typesetting packages
\usepackage{amsfonts, amsthm, amsmath, amssymb}
\usepackage{mathtools}  % Extension to amsmath

% Symbol and utility packages
\usepackage{cancel, textcomp}
\usepackage[mathscr]{euscript}
\usepackage[nointegrals]{wasysym}

% Extras
\usepackage{physics}  % Lots of useful shortcuts and macros
\usepackage{tikz-cd}  % For drawing commutative diagrams easily
\usepackage{color}  % Add some colour to life
\usepackage{microtype}  % Minature font tweaks

% Common shortcuts
\def\mbb#1{\mathbb{#1}}
\def\mfk#1{\mathfrak{#1}}

\def\bN{\mbb{N}}
\def\bC{\mbb{C}}
\def\bR{\mbb{R}}
\def\bQ{\mbb{Q}}
\def\bZ{\mbb{Z}}

% Sometimes helpful macros
\newcommand{\func}[3]{#1\colon#2\to#3}
\newcommand{\vfunc}[5]{\func{#1}{#2}{#3},\quad#4\longmapsto#5}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}

% Some standard theorem definitions
\newtheorem{Theorem}{Theorem}
\newtheorem{Proposition}{Theorem}
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Corollary}[Theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{Definition}[Theorem]{Definition}

% COPY AND PASTE BELOW TO USE FOR TEXIT
% more Blackboard letters
\def\bI{\mbb{I}}
\def\bH{\mbb{H}}

% brackets & parens stuff
\newcommand{\br}[1]{\left(#1\right)}
\newcommand{\sbr}[1]{\left[#1\right]}
\newcommand{\brc}[1]{\left\{#1\right\}}
\newcommand{\lbr}[1]{\left\langle#1\right\rangle}

% thanks Tuong for the \fxn cmd
\newcommand{\fxn}[5]{#1:\begin{array}{rcl}#2&\longrightarrow & #3\\[-0.5mm]#4&\longmapsto &#5\end{array}}
\newcommand{\sm}{\setminus}
\newcommand{\lc}{\epsilon_{ijk}}
\newcommand{\ep}{\epsilon}
\newcommand{\vp}{\varphi}
\newcommand{\sto}{\longrightarrow}
\newcommand{\cjg}[1]{\overline{#1}}
\newcommand{\ph}[2]{#1\angle\br{#2}}
\newcommand{\inv}{^{-1}}
\renewcommand{\L}{\mathcal{L}}
\newcommand{\F}{\mathcal{F}}

% more function names
\DeclareMathOperator{\arcsinh}{arcsinh}
\DeclareMathOperator{\arccosh}{arccosh}
\DeclareMathOperator{\arctanh}{arctanh}
\DeclareMathOperator{\arccsch}{arccsch}
\DeclareMathOperator{\arcsech}{arcsech}
\DeclareMathOperator{\arccoth}{arccoth}
\DeclareMathOperator{\cis}{cis}
\DeclareMathOperator{\sinc}{sinc}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\rect}{rect}
\DeclareMathOperator{\tri}{tri}

% linalg stuff
\renewcommand{\i}{\hat{\imath}}
\renewcommand{\j}{\hat{\jmath}}
\renewcommand{\k}{\hat{k}}
\renewcommand{\l}{\hat{\ell}}
\DeclareMathOperator{\cof}{cof}
\DeclareMathOperator{\adj}{adj}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\rref}{RREF}
\DeclareMathOperator{\REF}{REF}
\newcommand{\proj}[2]{\text{proj}_{#2}\br{#1}}
\newcommand{\m}[2][p]{\begin{#1matrix}#2\end{#1matrix}}
\newcommand{\arr}[3][\br]{#1{\begin{array}{#2}#3\end{array}}}

% for fun
\newcommand{\lhban}{~\color{red}{\cancel{\color{black}\fontsize{10}{0}\text{LH}}}\hspace{-16.6pt}\fontsize{60}{0}\raisebox{-2.7pt}\Circle}
\newcommand{\meow}{\approx\mkern-5mu\mathcal{O}\omega\mathcal{O}\mkern-5mu\approx}
\newcommand{\limdefn}{f'(x)=\lim_{h\to0}\tfrac{f(x+h)-f(x)}h}
\newcommand{\trig}{\text{A relatively small list of trigonometric identities:}\\\\\text{defining relations:}\\
\csc(x)=\tfrac1{\sin(x)}\quad\tan(x)=\tfrac{\sin(x)}{\cos(x)}\\
\sec(x)=\tfrac1{\cos(x)}\quad\cot(x)=\tfrac1{\tan(x)}=\tfrac{\cos(x)}{\sin(x)}\\\\
\text{pythagoras:}\\
\sin^2(x)+\cos^2(x)=1\\
1+\cot^2(x)=\csc^2(x)\\
\tan^2(x)+1=\sec^2(x)\\\\
\text{cofunction, negative angle, and supplement:}\\
\sin(x)=\cos(\tfrac\pi2-x)\quad\sin(-x)=-\sin(x)\\
\cos(x)=\sin(\tfrac\pi2-x)\quad\cos(-x)=\cos(x)\\
\tan(x)=\cot(\tfrac\pi2-x)\quad\tan(-x)=-\tan(x)\\
\sin(\pi-x)=\sin(x)\\
\cos(\pi-x)=-\cos(x)\\
\tan(\pi-x)=-\tan(x)\\\\
\text{periodicity - for all }n\in\bZ\text{, we have:}\\
\sin(x+2\pi n)=\sin(x)\\
\cos(x+2\pi n)=\cos(x)\\
\tan(x+\pi n)=\tan(x)\\\\
\text{addition:}\\
\sin(x\pm y)=\sin(x)\cos(y)\pm\cos(x)\sin(y)\\
\cos(x\pm y)=\cos(x)\cos(y)\mp\sin(x)\sin(y)\\
\tan(x\pm y)=\tfrac{\tan(x)\pm\tan(y)}{1\mp\tan(x)\tan(y)}\\\\
\text{double angle:}\\
\sin(2x)=2\sin(x)\cos(x)\\
\cos(2x)=\cos^2(x)-\sin^2(x)\\
\cos(2x)=2\cos^2(x)-1=1-2\sin^2(x)\\
\tan(2x)=\tfrac{2\tan(x)}{1-\tan^2(x)}\\\\
\text{sum-to-product and product-to-sum:}\\
\cos(x)+\cos(y)=2\cos(\tfrac{x+y}2)\cos(\tfrac{x-y}2)\\
\cos(x)-\cos(y)=-2\sin(\tfrac{x+y}2)\sin(\tfrac{x-y}2)\\
\sin(x)\pm\sin(y)=2\sin(\tfrac{x\pm y}2)\cos(\tfrac{x\mp y}2)\\
2\sin(x)\sin(y)=\cos(x-y)-\cos(x+y)\\
2\sin(x)\cos(y)=\sin(x+y)+\sin(x-y)\\
2\cos(x)\cos(y)=\cos(x+y)+\cos(x-y)\\\\
\text{Hope you find whatever identities you need!}}