Answer
We derive the given identities below.
Work Step by Step
We derive the identities as follows:
\begin{align*}
\tan (a+b)&=\frac{\sin (a+b)}{\cos (a+b)}\\
&=\frac{\sin a\cos b+\cos a\sin b}{ \cos a\cos b-\sin a\sin b}\\
&= \frac{\frac{\sin a\cos b}{ \cos a\cos b}+\frac{\cos a\sin b}{ \cos a\cos b}}{1-\frac{\sin a\sin b}{ \cos a\cos b}}\\
&= \frac{\tan a+\tan b}{1-\tan a \tan b}
\end{align*}
and
\begin{align*}
\cot(a-b)&=\frac{\cos (a-b)}{\sin (a-b)}\\
&=\frac{ \cos a\cos b+\sin a\sin b}{\sin a\cos b-\cos a\sin b}\\
&= \frac{\frac{ \cos a\cos b }{ \sin a\sin b }+\frac{\sin a\sin b}{\sin a\sin b }}{\frac{\sin a\cos b}{ \sin a\sin b}-\frac{\cos a\sin b}{ \sin a\sin b}}\\
&= \frac{\cot a \cot b+1}{\cot b-\cot a}
\end{align*}
