If tan $A = 1/7$ and tan $B = 1/3$ find tan($2A + B$) without using calculator approximations.

Step 1: Calculate tan(2A)

To find $\tan(2A+B)$, we first need to calculate $\tan(2A)$. We use the double angle formula for tangent, which states that $\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}$. We are given that $\tan A = \frac{1}{7}$.

Step 2: Substitute and Simplify tan(2A)

Now, substitute the value of $\tan A = \frac{1}{7}$ into the formula for $\tan(2A)$ and simplify the expression. This involves basic fraction arithmetic.

Step 3: Calculate tan($2A + B$)

Now that we have $\tan(2A)$ and are given $\tan B$, we can use the sum formula for tangent to find $\tan(2A+B)$. The formula is $\tan(X+Y) = \frac{\tan X + \tan Y}{1 - \tan X \tan Y}$. Here, $X = 2A$ and $Y = B$.

Step 4: Substitute and Simplify tan($2A + B$)

Substitute the calculated value of $\tan(2A) = \frac{7}{24}$ and the given value of $\tan B = \frac{1}{3}$ into the sum formula. Perform the necessary fraction additions, subtractions, and multiplications to simplify the expression to its final form.

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