If `cottheta=15/8`, then evaluate `((2+2sintheta)(1-sintheta))/((1+costheta)(2-2costheta))`

Options

A. `8/15`
B. `15/8`
C. `64/225`
D. `225/64`

Answer

To solve the problem step by step, we start with the given information:

**Given:**
\[ \cot \theta = \frac{15}{8} \]

### Step 1: Determine the values of sin θ and cos θ
Using the definition of cotangent, we can set up a right triangle where:
- The adjacent side (base) = 15
- The opposite side (perpendicular) = 8

To find the hypotenuse (h), we use the Pythagorean theorem:
\[ h = \sqrt{(15^2 + 8^2)} = \sqrt{225 + 64} = \sqrt{289} = 17 \]

Now we can find sin θ and cos θ:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{17} \]
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{15}{17} \]

### Step 2: Substitute sin θ and cos θ into the expression
We need to evaluate:
\[ \frac{(2 + 2 \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(2 - 2 \cos \theta)} \]

Substituting the values of sin θ and cos θ:
\[ = \frac{(2 + 2 \cdot \frac{8}{17})(1 - \frac{8}{17})}{(1 + \frac{15}{17})(2 - 2 \cdot \frac{15}{17})} \]

### Step 3: Simplify the numerator
Calculating the numerator:
\[ = (2 + \frac{16}{17})(1 - \frac{8}{17}) = \left(\frac{34}{17} + \frac{16}{17}\right)\left(\frac{9}{17}\right) = \frac{50}{17} \cdot \frac{9}{17} = \frac{450}{289} \]

### Step 4: Simplify the denominator
Calculating the denominator:
\[ = (1 + \frac{15}{17})(2 - 2 \cdot \frac{15}{17}) = \left(\frac{17}{17} + \frac{15}{17}\right)\left(\frac{34}{17} - \frac{30}{17}\right) = \frac{32}{17} \cdot \frac{4}{17} = \frac{128}{289} \]

### Step 5: Combine the results
Now we combine the results:
\[ \frac{\frac{450}{289}}{\frac{128}{289}} = \frac{450}{128} \]

### Step 6: Simplify the fraction
To simplify \(\frac{450}{128}\):
\[ = \frac{225}{64} \]

### Final Answer
Thus, the final answer is:
\[ \frac{225}{64} \]

---