Prove the identity: (sin θ + sin 2θ) / (1 + cos θ + cos 2θ) = tan θ

We will use the following identities:

• Double angle identity for sine: $\sin 2\theta = 2 \sin \theta \cos \theta$
• Double angle identity for cosine: $\cos 2\theta = 2 \cos^2 \theta - 1$
• Tangent identity: $\tan \theta = \frac{\sin \theta}{\cos \theta}$

The numerator is $\sin \theta + \sin 2\theta$.
Substitute $\sin 2\theta = 2 \sin \theta \cos \theta$:

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

Factor out $\sin \theta$:

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

The denominator is $1 + \cos \theta + \cos 2\theta$.
Substitute $\cos 2\theta = 2 \cos^2 \theta - 1$:

$$1 + \cos \theta + \cos 2\theta = 1 + \cos \theta + (2 \cos^2 \theta - 1)$$

Combine the constant terms ($1 - 1 = 0$):

$$= \cos \theta + 2 \cos^2 \theta$$

Factor out $\cos \theta$:

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

Given that $\cos \theta \neq -\frac{1}{2}$ and $\cos \theta \neq 0$, we know that $1 + 2 \cos \theta \neq 0$ and $\cos \theta \neq 0$. Therefore, we can cancel out the common term $(1 + 2 \cos \theta)$ from the numerator and denominator:

$$= \frac{\sin \theta}{\cos \theta}$$

Using the tangent identity:

$$= \tan \theta$$

Thus, the left-hand side simplifies to $\tan \theta$, which is equal to the right-hand side.
The identity is proven.