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### **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 \( x + y + z + 2xyz = 1 \).

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### **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 \( \cos \theta + \cos 2\theta + \cos 3\theta = 0 \).

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### **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. \( \lim_{x \to 2} \frac{\sqrt{2x} - \sqrt{3 - x^2}}{x - 2} \)

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### **19.**
**a)** Define the limit of a function.

**b)** Evaluate:
1. \( \lim_{x \to 0} x \cot \theta - \theta \cot x \), where \( x \to 0 \).

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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.

16(a) Prove that 4(cot 13°) + csc² 13° - √5 = π.
16(b) If cos x + cos y + cos z - 1 = π, show that x + y + z + 2xyz = 1.
17(a) If cos 2θ = cos α, prove that θ = nπ ± α/2.
17(b) Find the general value of cos θ + cos 2θ + cos 3θ = 0.
18(a) What do you mean by indeterminate form?
18(b) Evaluate the limits:
1. lim (x → 1) (x - 1)/(x - 1)
2. lim (x → 2) (x⁵ - 2⁵)/(x - 2)
3. lim (x → 2) (√(2x) - √(3 - x²))/(x - 2)
19(a) Define the limit of a function.
19(b) Evaluate: lim (x → 0) (x cotθ - θ cotx)

Explore solutions to advanced trigonometric proofs, limit evaluation problems, and the concept of indeterminate forms. Topics include proving trigonometric identities, solving complex limits, and understanding indeterminate forms. Suitable for Grades 11-12 students preparing for calculus.

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