It looks like you've uploaded a mathematical problem involving sequences and limits, specifically with trigonometric identities and convergence tests.

To address the steps from the file:

• (d) This part asks you to express $z_{n+1}$ in terms of $z_n$ using the Angle Sum identities for sine and cosine. The identities are:

  • $\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)$
  • $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$

  The goal is to write $z_{n+1} = (x_{n+1}, y_{n+1})$ as a function of $z_n = (x_n, y_n)$.

• (e) This part involves factoring out $\cos(\theta)$ from the prior work. Using the expression $\cos(\theta) = \frac{1}{\sqrt{1 + \tan^2(\theta)}}$, the task is to simplify your expressions.

• (f) In this section, you are asked to show that a common factor $K_n$ can be pulled out of both sequences, $z_{n+1}$ and $z_{n+1}$, and prove that the product converges to a non-zero value by applying a suitable convergence test.

Would you like me to break down any specific part of the solution or help with the calculations? Let me know how you'd like to proceed!