I see the uploaded image contains mathematical problems. Let me transcribe and explain each question step by step to assist you better.

16.

a) Prove that $4 (\cot 13^\circ) + \csc^2 13^\circ - \sqrt{5} = \pi$.

b) If $\cos x + \cos y + \cos z - 1 = \pi$, show that $x + y + z + 2xyz = 1$.

17.

a) If $\cos 2\theta = \cos \alpha$, prove that $\theta = n\pi \pm \frac{\alpha}{2}$, where $n$ is an integer.

b) Find the general value of $\cos \theta + \cos 2\theta + \cos 3\theta = 0$.

18.

a) What do you mean by indeterminate form?

b) Evaluate:

1. $\lim_{x \to 1} \frac{x - 1}{x - 1}$
2. $\lim_{x \to 2} \frac{x^5 - 2^5}{x - 2}$
3. $\lim_{x \to 2} \frac{\sqrt{2x} - \sqrt{3 - x^2}}{x - 2}$

19.

a) Define the limit of a function.

b) Evaluate:

1. $\lim_{x \to 0} x \cot \theta - \theta \cot x$, where $x \to 0$.

Would you like me to solve any specific problem or provide detailed explanations for all? Let me know your focus!

Here are five exploratory questions to expand this:

1. Can you clarify the concept of indeterminate forms for limits?
2. How does trigonometric proof utilize standard identities in proving such equations?
3. What is the significance of general solutions in trigonometric equations like $\cos \theta + \cos 2\theta + \cos 3\theta = 0$?
4. Can we apply L'Hôpital's Rule to solve any of these limits?
5. How do higher-order terms (like $x^5$) influence the calculation of limits?

Tip: Always remember to simplify trigonometric and algebraic expressions before attempting to solve equations or evaluate limits. This often avoids unnecessary complexity.