If tan(A + B) = p and tan(A − B) = q, Find tan 2B
Question:
If
\[
\tan(A+B)=p
\]
and
\[
\tan(A-B)=q
\]
find
\[
\tan2B
\]
Solution
\[ 2B=(A+B)-(A-B) \]
\[ \tan2B = \tan[(A+B)-(A-B)] \]
Using \[ \tan(C-D)=\frac{\tan C-\tan D}{1+\tan C\tan D} \]
\[ \tan2B = \frac{ \tan(A+B)-\tan(A-B) } { 1+\tan(A+B)\tan(A-B) } \]
\[ = \frac{p-q}{1+pq} \]
\[ \boxed{ \tan2B=\frac{p-q}{1+pq} } \]