Proof without words: trigonometric differentiation

Is this some kind of “proof without words”?

Visual trigonometric derivatives

As usual, you can download the PDF version.

Plotted with this ugly code in the following (using TikZ ):

\def\radiusa{6.0}
\def\radiusb{0.3}
\def\radiusc{5.2}
\def\radiusd{4.5}
\def\desa{1.4}
\def\desb{1.4}
\def\anglea{48}
\def\angleb{138}
\def\anglec{-42}
\def\angleap{55}
\begin{tikzpicture}[x=1.0cm,y=1.0cm,font=\sffamily]
  \clip (-4.0,-4.2) rectangle (24, 10);% 9 * 5
  % \draw[help lines,step=1] (-5,-5) grid (22,12);
  \draw [color=black!20,ultra thin] (-8, 0) -- (8, 0);
  \draw [color=black!20,ultra thin] (0, -8) -- (0, 8);
  \coordinate (O) at (0, 0);
  \coordinate (A) at (\anglea:\radiusa);
  \coordinate (Ap) at (\angleap:\radiusa);
  \coordinate (App) at (\radiusa * cos \angleap, \radiusa * sin \anglea);
  \coordinate (B) at (\radiusa * cos \anglea, 0);
  \coordinate (C) at (0, \radiusa * sin \anglea);
  \coordinate (D) at (\radiusa, 0);
  \coordinate (E) at (0, \radiusa);
  \coordinate (F) at (\radiusa, \radiusa * tan \anglea);
  \coordinate (Fp) at (\radiusa, \radiusa * tan \angleap);
  \coordinate (G) at (\radiusa * cot \anglea, \radiusa);
  \coordinate (Gp) at (\radiusa * cot \angleap, \radiusa);
  \coordinate (H) at (\radiusc * cos \anglec, \radiusc * sin \anglec);
  \coordinate (I) at (\radiusd * cos \angleb, \radiusd * sin \angleb);
  \coordinate (J) at (\radiusa * cot \anglea + \radiusd * cos \angleb, \radiusa + \radiusd * sin \angleb);
  \coordinate (K) at (\radiusa + \radiusc * cos \anglec, \radiusa * tan \anglea + \radiusc * sin \anglec);
  \coordinate (Ba) at (\radiusa * cos \anglea,\radiusb);
  \draw [color=black!20,ultra thick,dashed] (O) circle (\radiusa);
  \draw [color=black,thin] (0:\radiusb) arc(0:\anglea:\radiusb) node[right] {$\theta$};
  \draw [color=black,thick] (O) -- (A) node[midway,below,black] {1};
  \draw [color=red,thick] (A) -- (B) node[midway,right,red] {$\sin$};
  \draw [color=green!80!black,thick] (C) -- (A) node[midway,below,green!80!black] {$\cos$};
  \draw [color=magenta,thick] (D) -- (F) node[midway,right,magenta] {$\tan$};
  \draw [color=cyan!80!black,thick] (E) -- (G) node[midway,above,cyan!80!black] {$\cot$};
  \draw [color=blue!80!black,thick] (O) -- (G);
  \draw [color=orange,thick] (O) -- (F);
  \draw [color=blue!80!black,thick,dashed] (O) -- (I);
  \draw [color=blue!80!black,thick,dashed,name path=lgj] (G) -- (J);
  \draw [color=orange,thick,dashed] (O) -- (H);
  \draw [color=orange,thick,dashed] (F) -- (K);
  \draw [color=white,ultra thin,name path=lfk] (F) -- ++(\angleb:\radiusc);
  \draw [color=blue!80!black,thick,stealth-stealth] (I) -- (J) node[sloped,midway,above,blue!80!black] {$\csc$};
  \draw [color=orange,thick,stealth-stealth] (H) -- (K) node[sloped,midway,below,orange] {$\sec$};
  \draw [color=black,thin] (Ba) -- ++(180:\radiusb) -- ++(270:\radiusb);
  % theta + \delta theta
  \draw [color=black,dashed,name path=lofp] (O) -- (Fp);
  \draw [color=black,ultra thick,fill=red!60] (A)
  -- (Ap) node[midway,above right=-3pt] {$d\theta$}
  -- (App) -- cycle;
  \fill [name intersections={of=lofp and lgj,name=Gpp}];
  \draw [color=black,ultra thick,fill=cyan!50] (G) -- (Gp)
  -- (Gpp-1) -- cycle node[midway,above right=-5pt] {$df$};
  \fill [name intersections={of=lofp and lfk,name=Fpp}];
  \draw [color=black,ultra thick,fill=magenta!50] (F) -- (Fp)
  -- (Fpp-1) -- cycle;
  \node at (5.65, 7.25) {$dg$};
  \draw [color=black!50,thick,dashed] (11.6, -4) -- (11.6, 9);
  % new triangle 1
  \coordinate (NApp) at ($(13.4, -1)$);
  \coordinate (NAp)  at ($(Ap) - (App) + (NApp)$);
  \coordinate (NA)   at ($(A) -(App) + (NApp)$);
  \draw [color=black,ultra thick,fill=red!60] (NA) 
  -- (NAp) node[midway,above right=-2pt] {$d\theta$}
  -- (NApp) node[midway,left] {$d\sin$}
  -- cycle node[midway,below] {$d\cos$};
  \coordinate (NApt) at ($(NAp) + ( \radiusb * sin \anglea, - \radiusb * cos \anglea)$);
  \draw [color=black,thin] (NApt) arc(\anglec:-90:\radiusb);
  \draw [color=black!50,thick,dashed] (12, 1) -- (24, 1);
  % new trangle 2
  \coordinate (NGp)  at ($(13.0, 3)$);
  \coordinate (NG)   at ($(G) - (Gp) + (NGp) $);
  \coordinate (NGpp) at ($(Gpp-1) - (Gp) + (NGp)$);
  \draw [color=black,ultra thick,fill=cyan!50] (NG) 
  -- (NGp)  node[midway,below] {$d\cot$}
  -- (NGpp) node[midway,above left=-2pt] {$d\csc$}
  -- cycle  node[midway,above right=-3pt] {$df$};
  \coordinate (NGpt) at ($(NGp) + (\radiusb, 0)$);
  \draw [color=black,thin] (NGpt) arc(0:\anglea:\radiusb);
  \draw [color=black!50,thick,dashed] (12, 5) -- (24, 5);
  % new trangle 3
  \coordinate (NFpp) at ($(13.1, 7)$);
  \coordinate (NFp)  at ($(Fp) - (Fpp-1) + (NFpp)$);
  \coordinate (NF)   at ($(F) - (Fpp-1) + (NFpp)$);
  \draw [color=black,ultra thick,fill=magenta!50] (NF)
  -- (NFp)  node[midway,right] {$d\tan$}
  -- (NFpp) node[midway,above left] {$d\sec$}
  -- cycle  node[midway,below left=-3pt] {$dg$};
  \coordinate (NFt) at ($(NF) + (0, \radiusb)$);
  \draw [color=black,thin] (NFt) arc(90:\angleb:\radiusb);
  % NOTES
  \fill (6.6, 8) circle (2pt) node [right] {\large $\displaystyle \frac{d\theta}{1} = \frac{df}{\csc \theta} = \frac{dg}{\sec \theta}$};
  \fill (16, -1.5) circle (2pt) node [right] {\large $d\cos\theta = -\sin\theta d\theta$};
  \fill (16, -0.5) circle (2pt) node [right] {\large $d\sin\theta = \cos\theta d\theta$};
  \fill (16, 3.8) circle (2pt) node [right] {\large $\displaystyle d\cot\theta = -\frac{df}{\sin\theta} = -\csc^2\theta d\theta$};
  \fill (16, 2.5) circle (2pt) node [right] {\large $\displaystyle d\csc\theta = -\frac{df}{\tan \theta} = -\csc\theta \cot\theta d\theta$};
  \fill (16, 6.5) circle (2pt) node [right] {\large $\displaystyle d\sec\theta = \tan\theta dg = \sec\theta \tan\theta d\theta$};
  \fill (16, 7.8) circle (2pt) node [right] {\large $\displaystyle d\tan\theta = \frac{dg}{\cos \theta} = \sec^2\theta d\theta$};
\end{tikzpicture}

Happy $\LaTeX$ing!

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