To find the reciprocal of \(\frac{\frac{1}{3}}{\left(\frac{1}{3} + \frac{1}{4}\right)}\), follow these steps:

1. Calculate the denominator:

\[
\frac{1}{3} + \frac{1}{4}
\]

To add these fractions, find a common denominator. The least common denominator for 3 and 4 is 12.

\[
\frac{1}{3} = \frac{4}{12}
\]
\[
\frac{1}{4} = \frac{3}{12}
\]

Adding these gives:

\[
\frac{4}{12} + \frac{3}{12} = \frac{7}{12}
\]

2. Form the overall fraction:

\[
\frac{\frac{1}{3}}{\frac{7}{12}}
\]

Dividing by a fraction is the same as multiplying by its reciprocal:

\[
\frac{1}{3} \times \frac{12}{7} = \frac{12}{21} = \frac{4}{7}
\]

3. Find the reciprocal:

The reciprocal of \(\frac{4}{7}\) is \(\frac{7}{4}\).

The correct answer is: A. \(\frac{7}{4}\)