[tex]\bf cos(\alpha+\beta)cos(\alpha-\beta)\\\\
-----------------------------\\\\
\textit{using the sum identities}\\\\\
\begin{array}{ccccccll}
[cos(\alpha)cos(\beta)&-&sin(\alpha)sin(\beta)]&[cos(\alpha)cos(\beta)&+&sin(\alpha)sin(\beta)]\\
a&-&b&a&+&b
\end{array}\\\\
\textit{notice above, is just a difference of squares, thus}
\\\\\\\
[cos(\alpha)cos(\beta)]^2-[sin(\alpha)sin(\beta)]^2\\\\
\textit{now let us distribute the exponents}\\\\[/tex]
[tex]\bf cos^2(\alpha)cos^2(\beta)-sin^2(\alpha)sin^2(\beta)\\\\
-----------------------------\\\\
\textit{now, let us use the pythagorean identity of }\\\\
sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta)\\\\
-----------------------------\\\\\
[ 1-sin^2(\alpha)][ 1-sin^2(\beta)]-sin^2(\alpha)sin^2(\beta)
\\\\\\\
1-sin^2(\beta)-sin^2(\alpha)\underline{+sin^2(\alpha)sin^2(\beta)-sin^2(\alpha)sin^2(\beta)}
\\\\\\
\underline{1-sin^2(\beta)}-sin^2(\alpha)\implies cos^2(\beta)-sin^2(\alpha)[/tex]
