To solve question 16, we need to evaluate the expression given:

$\text{If } 8 \cot \theta = 15, \text{ find the value of } \frac{(2 + 3 \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(2 - 2 \cos \theta)}.$

Step 1: Express $\cot \theta$ in terms of $\sin \theta$ and $\cos \theta$

Given: $8 \cot \theta = 15 \Rightarrow \cot \theta = \frac{15}{8}$ We know that: $\cot \theta = \frac{\cos \theta}{\sin \theta} \Rightarrow \frac{\cos \theta}{\sin \theta} = \frac{15}{8}$

Step 2: Use the identity $\sin^2 \theta + \cos^2 \theta = 1$

Let $\sin \theta = x$ and $\cos \theta = y$. Then: $\frac{y}{x} = \frac{15}{8} \Rightarrow y = \frac{15}{8}x$

Substituting into the Pythagorean identity: $x^2 + \left(\frac{15}{8}x\right)^2 = 1$ $x^2 + \frac{225}{64}x^2 = 1$ $x^2\left(1 + \frac{225}{64}\right) = 1$ $x^2 \cdot \frac{289}{64} = 1 \Rightarrow x^2 = \frac{64}{289} \Rightarrow x = \frac{8}{17}$

So, $\sin \theta = \frac{8}{17}$ and $\cos \theta = \frac{15}{17}$

Step 3: Substitute $\sin \theta$ and $\cos \theta$ into the expression

Now we substitute these into the given expression: $\frac{(2 + 3 \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(2 - 2 \cos \theta)}$

Substituting $\sin \theta = \frac{8}{17}$ and $\cos \theta = \frac{15}{17}$: $\frac{\left(2 + 3 \cdot \frac{8}{17}\right)\left(1 - \frac{8}{17}\right)}{\left(1 + \frac{15}{17}\right)\left(2 - 2 \cdot \frac{15}{17}\right)}$

Step 4: Simplify each part

1. Simplify the numerator: $2 + \frac{24}{17} = \frac{34 + 24}{17} = \frac{58}{17}$ $1 - \frac{8}{17} = \frac{17 - 8}{17} = \frac{9}{17}$ So, the numerator is: $\frac{58}{17} \times \frac{9}{17} = \frac{522}{289}$

2. Simplify the denominator: $1 + \frac{15}{17} = \frac{32}{17}$ $2 - \frac{30}{17} = \frac{34 - 30}{17} = \frac{4}{17}$ So, the denominator is: $\frac{32}{17} \times \frac{4}{17} = \frac{128}{289}$

Step 5: Calculate the final value

Now, divide the numerator by the denominator: $\frac{\frac{522}{289}}{\frac{128}{289}} = \frac{522}{128} = \frac{261}{64}$

So, the final answer is: $\frac{261}{64}$

Would you like more details or have any questions? Here are five related questions you might find helpful:

1. How do you solve problems involving trigonometric identities?
2. What are the key steps in simplifying complex trigonometric expressions?
3. How does the Pythagorean identity help in solving trigonometric equations?
4. What other identities are useful in manipulating trigonometric expressions?
5. How can you check your work when solving trigonometric equations?

Tip: Always ensure to simplify expressions fully before substituting values to avoid unnecessary complexity in calculations.