Q.If $\tan A=\dfrac12$ and $\tan B=\dfrac13$, find the value of $\tan(A+B)$.

The tangent compound-angle formula lets $\tan(A+B)$ be found directly from $\tan A$ and $\tan B$ without needing the individual angles at all.

Step 1 — write down the formula. $\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}$.

Step 2 — compute the numerator. $\tan A+\tan B=\dfrac12+\dfrac13=\dfrac{3+2}{6}=\dfrac56$.

Step 3 — compute the denominator. $1-\tan A\tan B=1-\dfrac12\cdot\dfrac13=1-\dfrac16=\dfrac56$.

Step 4 — divide. $\tan(A+B)=\dfrac{5/6}{5/6}=1$. …

Forgetting the $1-\tan A\tan B$ in the denominator and simply adding $\tan A+\tan B$ as the final answer; also mixing up the $+$ and …