Simplify the expression ( theta + 2theta)/(1 + theta + 2theta).

Step 2: Simplify the numerator of the LHS.
The numerator is the top part of the fraction, which is $\sin \theta + \sin 2\theta$.

To simplify this, we use a double angle identity for sine. The identity states that $\sin 2\theta$ can be rewritten as $2 \sin \theta \cos \theta$. This substitution is crucial because it allows us to express both terms in the numerator using $\sin \theta$ and $\cos \theta$ of a single angle $\theta$, rather than having terms with $2\theta$.

Substituting $\sin 2\theta = 2 \sin \theta \cos \theta$ into the numerator gives:

$$\sin \theta + \sin 2\theta = \sin \theta + 2 \sin \theta \cos \theta$$

Now, we observe that both terms on the right-hand side, $\sin \theta$ and $2 \sin \theta \cos \theta$, share a common factor of $\sin \theta$.

To simplify further, we factor out this common term $\sin \theta$:

$$= \sin \theta (1 + 2 \cos \theta)$$

When $\sin \theta$ is factored out from $\sin \theta$, it leaves 1 (since $\sin \theta = \sin \theta \cdot 1$). When $\sin \theta$ is factored out from $2 \sin \theta \cos \theta$, it leaves $2 \cos \theta$. This results in the simplified form of the numerator: $\sin \theta (1 + 2 \cos \theta)$. This factored form is useful because it often allows for cancellation with terms in the denominator, as seen in the full solution.