If `cotalpha=(15)/(8)`, then `((2+2sinalpha)(1-sinalpha))/((1+cosalpha)(2-2cosalpha))=?`

Options

A. `(125)/(8)`
B. `(225)/(64)`
C. `(64)/(225)`
D. `(8)/(125)`

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{(2 + 2\sin\alpha)(1 - \sin\alpha)}{(1 + \cos\alpha)(2 - 2\cos\alpha)} \]

given that \(\cot\alpha = \frac{15}{8}\).

### Step 1: Determine \(\sin\alpha\) and \(\cos\alpha\)

Since \(\cot\alpha = \frac{\cos\alpha}{\sin\alpha} = \frac{15}{8}\), we can set:
- \(\cos\alpha = 15k\)
- \(\sin\alpha = 8k\)

for some \(k > 0\).

Using the Pythagorean identity \(\sin^2\alpha + \cos^2\alpha = 1\):
\[ (8k)^2 + (15k)^2 = 1 \]
\[ 64k^2 + 225k^2 = 1 \]
\[ 289k^2 = 1 \]
\[ k^2 = \frac{1}{289} \implies k = \frac{1}{17} \]

Now substituting \(k\) back to find \(\sin\alpha\) and \(\cos\alpha\):
\[ \sin\alpha = 8k = \frac{8}{17}, \quad \cos\alpha = 15k = \frac{15}{17} \]

### Step 2: Substitute \(\sin\alpha\) and \(\cos\alpha\) into the expression

Now we substitute \(\sin\alpha\) and \(\cos\alpha\) into the original expression:

\[ \frac{(2 + 2\cdot\frac{8}{17})(1 - \frac{8}{17})}{(1 + \frac{15}{17})(2 - 2\cdot\frac{15}{17})} \]

### Step 3: Simplify the numerator

Calculating the numerator:
\[ 2 + 2\cdot\frac{8}{17} = 2 + \frac{16}{17} = \frac{34}{17} + \frac{16}{17} = \frac{50}{17} \]
\[ 1 - \frac{8}{17} = \frac{17}{17} - \frac{8}{17} = \frac{9}{17} \]

Thus, the numerator becomes:
\[ \frac{50}{17} \cdot \frac{9}{17} = \frac{450}{289} \]

### Step 4: Simplify the denominator

Calculating the denominator:
\[ 1 + \frac{15}{17} = \frac{17}{17} + \frac{15}{17} = \frac{32}{17} \]
\[ 2 - 2\cdot\frac{15}{17} = 2 - \frac{30}{17} = \frac{34}{17} - \frac{30}{17} = \frac{4}{17} \]

Thus, the denominator becomes:
\[ \frac{32}{17} \cdot \frac{4}{17} = \frac{128}{289} \]

### Step 5: Combine the results

Now we can combine the results:
\[ \frac{\frac{450}{289}}{\frac{128}{289}} = \frac{450}{128} \]

### Step 6: Simplify the fraction

We can simplify \(\frac{450}{128}\):
\[ \frac{450 \div 2}{128 \div 2} = \frac{225}{64} \]

### Final Answer

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