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

\renewcommand{\div}{\divisionsymbol}


\newcommand{\y}{\infty}
\newcommand{\trigsum}{
\begin{itemize}
    \item $\sin(x+y) = \sin x \cos y + \cos x \sin y$
    \item $\sin(x-y) = \sin x \cos y - \cos x \sin y$
    \item $\cos(x+y) = \cos x \cos y - \sin x \sin y$
    \item $\cos(x-y) = \cos x \cos y - \sin x \sin y$
    \item $\tan (x+y) = \frac{\tan x + \tan y}{1- \tan x \tan y}$
    \item $\tan (x-y) = \frac{\tan x - \tan y}{1+ \tan x \tan y}$
    \item $\cot (x+y) = \frac{\cot y \cot x - 1}{\cot y + \cot x}$
    \item $\cot (x-y) = \frac{\cot y \cot x + 1}{\cot y - \cot x}$
    
\end{itemize}
}
\newcommand{\trigsumprod}{
\begin{itemize}
    \item $\sin x + \sin y = 2 \sin \left (\frac{x+y}{2}\right) \cos \left (\frac{x-y}{2} \right)$
    \item $\sin x - \sin y = 2 \sin \left (\frac{x-y}{2}\right) \cos \left (\frac{x+y}{2} \right)$
    \item $\cos x + \cos y = 2 \cos \left (\frac{x+y}{2}\right) \cos \left (\frac{x-y}{2} \right)$
    \item $\cos x + \cos y = 2 \sin \left (\frac{x+y}{2}\right) \cos \left (\frac{x-y}{2} \right)$
\end{itemize}
}

\newcommand{\trigangles}{
\begin{itemize}
    \item $\sin (2 x) = 2 \sin x \cos x $
    \item $\cos (2x) = \cos ^2 x - \sin ^2 x \ = \ 1-2 \sin ^2 x  \ = \ 2 \cos ^2 x -1$
    \item $\tan (2x) = \frac{2 \tan x}{1- \tan ^2 x}$
    \item $\sin (3x)= 3 \sin x - 4\sin^3 x$
    \item $\cos(3x) = 4\cos^3x-3\cos x$
\end{itemize}
}
\newcommand{\trigid}{
\begin{itemize}
    \item $\sin ^2 a + \cos ^2 a = 1$
    \item $1+ \tan ^2 a = \sec ^2 a$
    \item $1+ \cot ^2 a= \csc ^2 a$
\end{itemize}
}
\definecolor{hmm}{HTML}{F9DF55}
\definecolor{sky}{HTML}{66ffff}
\newcommand{\sus}{

\pagecolor{hmm}
\begin{tikzpicture} [scale=0.3]
    \draw [line width=0.5mm, black , fill=red ]
(0,5) to[bend right=30] 
(-0.5,4) to[bend right=30]
 (0,3) to[bend right=5] 
(0,0)to [bend right = 30]
(0.6,-1) to[bend right=30] 
(1.5,-1) to[bend right=30] (1.8,0) 
(0,5) to[bend left=45]  
(2,6.5) to[bend left=45] 
(4,4) --
(4,0) to[bend left=30] 
(3.5,-1) to[bend left=30] 
(2.5,-1) to[bend left=20]
(2.2,0) to[bend left=25](1.8,0)
(4,4.5) to[bend left=30]
(5,3.5) --
(5,2) to[bend left=45] (4,1);
\draw [line width=0.5mm, black , fill=sky,]
(2,5)--(0,5) to[bend right=30] (-0.5,4)  to[bend right=30] (0,3)
(2,5) to[bend left=30] (2.5,4) to[bend left=30] (2,3)--(0,3);
\draw [line width = 0.5mm, black]
(2.2,0) to [bend left = 25] (1.8,0) to [bend left = 5] (1.65,0.08)
(2.2,0) to [ bend left =8] (2.3,0.2); 
\end{tikzpicture}
}
\newcommand {\meow}{

\begin{tikzpicture}[scale=0.3]
\draw[ultra thick]
(-
1,6) to 
(1,6) to [bend left =3] 
(2.6,6.6) to [bend left =7]  
(3.1,6.72) to [bend left = 50]
(3.4,6.4) to [bend left=5]
(3.28,5.5) to [bend left = 2]
(2.98,4.45) to [bend left = 40]
(2.98,1.7) to [bend left = 50]
(-2.98,1.7) to [bend left=40]
(-2.98,4.45) to [bend left=2]
(-3.28,5.5) to [bend left=5]
(-3.4,6.4) to [bend left=50]
(-3.1,6.72) to [bend left =7] 
(-2.6,6.6) to [bend left =3]
(-1,6);
\draw (-2.5,2) to[bend right =10] (-4,1.5);
\draw (-2.5,2.5) to [bend right =10] (-4.2,2.2);
\draw (-2.5,3) to[bend right =10] (-4.3,2.9);
\draw (2.5,2) to[bend left =10] (4,1.5);
\draw (2.5,2.5) to [bend left =10] (4.2,2.2);
\draw (2.5,3) to[bend left =10] (4.3,2.9);
\end{tikzpicture}
}

\newcommand{\meowmeow}{
\begin{tikzpicture} [scale=0.3]
\draw[ultra thick]
(-1,6) to 
(1,6) to [bend left =3] 
(2.6,6.6) to [bend left =7]  
(3.1,6.72) to [bend left = 50]
(3.4,6.4) to [bend left=5]
(3.28,5.5) to [bend left = 2]
(2.98,4.45) to [bend left = 40]
(2.98,1.7) to [bend left = 5]
(2.25,0.9) to [bend right =4]
(2.75,-0.5) to [bend left =3]
(3.5,-2) to [bend left = 10]
(3.75,-4) to [bend left ]
(2.2,-6) to [bend left = 4]
(1,-6.3) to [bend left=4]
(-1,-6.3) to [bend left =4]
(-2.2,-6) to [bend left]
(-3.75,-4) to [bend left=10]
(-3.5,-2) to [bend left =3 ]
(-2.75,-0.5) to  [bend right = 4]
(-2.25,0.9) to [bend left =5]
(-2.98,1.7) to [bend left=40]
(-2.98,4.45) to [bend left=2]
(-3.28,5.5) to [bend left=5]
(-3.4,6.4) to [bend left=50]
(-3.1,6.72) to [bend left =7] 
(-2.6,6.6) to [bend left =3]
(-1,6);
\draw[ultra thick]
(-3.75,-4)to[bend left]
(-5,0) to [bend left=40]
(-4.5,0.3) to [bend left]
(-4.3,-0.2) to [bend right]
(-3.75,-2.95);
\draw (-2.5,2) to[bend right =10] (-4,1.5);
\draw (-2.5,2.5) to [bend right =10] (-4.2,2.2);
\draw (-2.5,3) to[bend right =10] (-4.3,2.9);
\draw (2.5,2) to[bend left =10] (4,1.5);
\draw (2.5,2.5) to [bend left =10] (4.2,2.2);
\draw (2.5,3) to[bend left =10] (4.3,2.9);
\end{tikzpicture}
}

%\definecolor{page}{HTML}{949397} %
%\definecolor{text}{HTML}{F4DF4E} % 
%\pagecolor{page}
%\begingroup \def\begin#1{\endgroup \begin{#1} \color{text}} 


\pgfmathsetmacro{\R}{\pdfuniformdeviate 256}
\pgfmathsetmacro{\G}{\pdfuniformdeviate 256}
\pgfmathsetmacro{\B}{\pdfuniformdeviate 256}

\definecolor{randomcolor}{RGB}{\R, \G,\B}
  \color{randomcolor}

\newcommand\Rp[1]{\the\numexpr#1+255\relax} 
\newcommand\Gp[1]{\the\numexpr#1+255\relax} 
\newcommand\Bp[1]{\the\numexpr#1+255\relax}

\definecolor{ppg}{RGB}{\Rp{-\R},\Gp{-\G}, \Bp{-\B}
}
\pagecolor{ppg}