Solving a Physics Problem: Sine vs. Cosine Theta

[Faiq]

Homework Statement

Question is provided

The Attempt at a Solution

The required solution is provided

My question is, the examiner have used sine theta in their expression for change in potential. I keep ending up with 1 - cosine theta
Second question is how did they get rid of theta in the proceeding equation

 

[mfb]

See the left side. For small angles, $\sin \theta \approx \theta$ and $\cos \theta \approx 1 - \frac 1 2 \theta^2$ (sometimes $\cos \theta \approx 1$ is sufficient).