# LaTeX2HTML 2022 (Released January 1, 2022)
# Associate images original text with physical files.
$key = q/displaystyledeleqno;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img10.png"
 ALT="$\displaystyle \deleqno$">|; 
$key = q/displaystylenydeleqno;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img11.png"
 ALT="$\displaystyle \nydeleqno$">|; 
$key = q/displaystylerdeqno;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img14.png"
 ALT="$\displaystyle \rdeqno$">|; 
$key = q/displaystylereqno;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img8.png"
 ALT="$\displaystyle \reqno$">|; 
$key = q/displaystylerndeqno;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img12.png"
 ALT="$\displaystyle \rndeqno$">|; 
$key = q/displaystyle{frac{{1-cos^2alpha}}{{cos^2alpha}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img15.png"
 ALT="$\displaystyle {\frac{{1 - \cos ^2 \alpha}}{{\cos ^2 \alpha}}}$">|; 
$key = q/displaystyle{frac{{2tanalpha}}{{1-tan^2alpha}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img9.png"
 ALT="$\displaystyle {\frac{{2\tan \alpha}}{{1 - \tan ^2 \alpha}}}$">|; 
$key = q/displaystyle{frac{{sin^2alpha}}{{1-sin^2alpha}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img13.png"
 ALT="$\displaystyle {\frac{{\sin ^2 \alpha}}{{1 - \sin ^2 \alpha}}}$">|; 
$key = q/displaystyle{frac{{sin^2alpha}}{{cos^2alpha}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img6.png"
 ALT="$\displaystyle {\frac{{\sin ^2 \alpha}}{{\cos ^2 \alpha}}}$">|; 
$key = q/displaystyle{frac{{sinalpha}}{{cosalpha}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img7.png"
 ALT="$\displaystyle {\frac{{\sin \alpha}}{{\cos \alpha}}}$">|; 
$key = q/mathcal{END};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img17.png"
 ALT="$\mathcal {END}$">|; 
$key = q/{ddeqar}cos(alpha+beta)&=&cosalphacosbeta-sinalphasinbetacos(alpha-beta)&=&cosalphacosbeta+sinalphasinbeta{ddeqar};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
  STYLE="" SRC="|."$dir".q|img4.png"
 ALT="\begin{ddeqar}
\cos ( \alpha + \beta ) &amp; =
&amp; \cos \alpha \cos \beta - \sin \al...
...a - \beta ) &amp; =
&amp; \cos \alpha \cos \beta + \sin \alpha \sin \beta
\end{ddeqar}">|; 
$key = q/{ddeqar}sin2alpha&=&2sinalphacosalphaheqnocos2alpha&=&cos^2alpha-sin^2alphanydeqnorem{or}&=&2cos^2alpha-1{ddeqar};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
  STYLE="" SRC="|."$dir".q|img5.png"
 ALT="\begin{ddeqar}
\sin 2\alpha &amp; =
&amp; 2 \sin \alpha \cos \alpha \heqno
\cos 2...
...- \sin ^2 \alpha \nydeqno \\\\
\rem{or}
&amp; = &amp; 2\cos ^2 \alpha - 1
\end{ddeqar}">|; 
$key = q/{ddeqn}cos(-alpha)=cosalpha{ddeqn};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img2.png"
 ALT="\begin{ddeqn}
\cos (-\alpha) = \cos \alpha
\end{ddeqn}">|; 
$key = q/{deqarr}cotalpha&=&frac{cosalpha}{sinalpha}where&=&frac{1}{tanalpha}{deqarr};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
  STYLE="" SRC="|."$dir".q|img16.png"
 ALT="\begin{deqarr}
\cot \alpha &amp; = &amp; \frac{\cos \alpha}{\sin \alpha} \\\\
\where
&amp; = &amp; \frac{1}{\tan \alpha}
\end{deqarr}">|; 
$key = q/{deqarr}sin(alpha+beta)&=&sinalphacosbeta+sinbetacosalphasin(alpha-beta)&=&sinalphacosbeta-sinbetacosalpha{deqarr};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
  STYLE="" SRC="|."$dir".q|img3.png"
 ALT="\begin{deqarr}
\sin ( \alpha + \beta ) &amp; =
&amp; \sin \alpha \cos \beta + \sin \be...
...a - \beta ) &amp; =
&amp; \sin \alpha \cos \beta - \sin \beta \cos \alpha
\end{deqarr}">|; 
$key = q/{deqn}sin(-alpha)=-sinalpha{deqn};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="" SRC="|."$dir".q|img1.png"
 ALT="\begin{deqn}
\sin (-\alpha) = - \sin \alpha
\end{deqn}">|;