Derivation of the tangent half angle identity

Hint: The numerator can be written as $$ \sin\theta = \sin \left(2 \cdot \frac{\theta}{2}\right) = 2\sin\frac{\theta}{2}\cos\frac{\theta}{2}. $$

It is a lot easier to use some trig. identities: $\sin (\theta) =2 \sin (\theta /2) \cos (\theta /2)$ and $1+\cos \theta =2\cos ^{2} (\theta /2)$. You will immediately get the result from these two formulas.

It's possible to understand many trigonometric identities with the unit circle.

Once you understand them, it's also much easier to remember them.

You can use two unit circles (one for $a$, one for $b$) to build a rhombus:

The sides of this rhombus have length 1. From this diagram, you can see that:

$$\tan\left(\frac{a+b}{2}\right) = \frac{\sin\left(\frac{a+b}{2}\right)}{\cos\left(\frac{a+b}{2}\right)}=\frac{\sin\left(a\right) + \sin\left(b\right)}{\cos\left(a\right)+\cos\left(b\right)}$$

In particular, with $a = 0$ and $b = \theta$, you have:

$$\tan\left(\frac{\theta}{2}\right) = \frac{0 + \sin\left(\theta\right)}{1 + \cos\left(\theta\right)}$$