If tan(A+B) = x and tan(A−B) = y, Find tan2A and tan2B

Question

If

\[ \tan(A+B)=x \]

and

\[ \tan(A-B)=y \]

find the values of:

\[ \tan2A \quad \text{and} \quad \tan2B \]

Solution

Since,

\[ 2A=(A+B)+(A-B) \]

\[ \tan2A = \tan[(A+B)+(A-B)] \]

\[ = \frac{\tan(A+B)+\tan(A-B)} {1-\tan(A+B)\tan(A-B)} \]

\[ = \frac{x+y}{1-xy} \]

Therefore,

\[ \boxed{\tan2A=\frac{x+y}{1-xy}} \]

Also,

\[ 2B=(A+B)-(A-B) \]

\[ \tan2B = \tan[(A+B)-(A-B)] \]

\[ = \frac{\tan(A+B)-\tan(A-B)} {1+\tan(A+B)\tan(A-B)} \]

\[ = \frac{x-y}{1+xy} \]

Therefore,

\[ \boxed{\tan2B=\frac{x-y}{1+xy}} \]