$\frac{\sin 10^\circ + \sin 20^\circ}{\cos 10^\circ + \cos 20^\circ}$ equals

A. $\tan 10^\circ + \tan 20^\circ$
B. $\tan 30^\circ$
C. $\frac{1}{2} (\tan 10^\circ + \tan 20^\circ)$
D. $\tan 15^\circ$
E. $\frac{1}{4} \tan 60^\circ$
1. We start by using the angle addition and subtraction formulas for sine and cosine to express $\sin 10^\circ$, $\sin 20^\circ$, $\cos 10^\circ$, and $\cos 20^\circ$ in terms of $\sin 15^\circ$, $\cos 15^\circ$, $\sin 5^\circ$, and $\cos 5^\circ$.

   \[
   \sin 10^\circ = \sin (15^\circ - 5^\circ) = \sin 15^\circ \cos 5^\circ - \cos 15^\circ \sin 5^\circ
   \]

   \[
   \sin 20^\circ = \sin (15^\circ + 5^\circ) = \sin 15^\circ \cos 5^\circ + \cos 15^\circ \sin 5^\circ
   \]

   \[
   \cos 10^\circ = \cos (15^\circ - 5^\circ) = \cos 15^\circ \cos 5^\circ + \sin 15^\circ \sin 5^\circ
   \]

   \[
   \cos 20^\circ = \cos (15^\circ + 5^\circ) = \cos 15^\circ \cos 5^\circ - \sin 15^\circ \sin 5^\circ
   \]

2. Next, we add the expressions for $\sin 10^\circ$ and $\sin 20^\circ$:

   \[
   \sin 10^\circ + \sin 20^\circ = (\sin 15^\circ \cos 5^\circ - \cos 15^\circ \sin 5^\circ) + (\sin 15^\circ \cos 5^\circ + \cos 15^\circ \sin 5^\circ)
   \]

   Simplifying, we get:

   \[
   \sin 10^\circ + \sin 20^\circ = 2 \sin 15^\circ \cos 5^\circ
   \]

3. Similarly, we add the expressions for $\cos 10^\circ$ and $\cos 20^\circ$:

   \[
   \cos 10^\circ + \cos 20^\circ = (\cos 15^\circ \cos 5^\circ + \sin 15^\circ \sin 5^\circ) + (\cos 15^\circ \cos 5^\circ - \sin 15^\circ \sin 5^\circ)
   \]

   Simplifying, we get:

   \[
   \cos 10^\circ + \cos 20^\circ = 2 \cos 15^\circ \cos 5^\circ
   \]

4. Now, we can form the ratio:

   \[
   \frac{\sin 10^\circ + \sin 20^\circ}{\cos 10^\circ + \cos 20^\circ} = \frac{2 \sin 15^\circ \cos 5^\circ}{2 \cos 15^\circ \cos 5^\circ}
   \]

   Simplifying, we get:

   \[
   \frac{\sin 10^\circ + \sin 20^\circ}{\cos 10^\circ + \cos 20^\circ} = \frac{\sin 15^\circ}{\cos 15^\circ} = \tan 15^\circ
   \]

5. Therefore, the expression simplifies to $\tan 15^\circ$.

The final answer is $\boxed{\tan 15^\circ}$