`S_(10) = cos'(pi)/(180) + cos'(3pi)/(180) + "….." cos '(19pi)/(180)`

Options

A. `(sin'pi/9cos'pi/18)/(sin'pi/90)`
B. `(sin'pi/18cos'pi/18)/(sin(pi/180))`
C. `(sin'pi/18cos'pi/9)/(sin(pi/180))`
D. `(cos'pi/18cos'pi/9)/(sin(pi/180))`

Answer

To solve the problem \( S_{10} = \cos\left(\frac{\pi}{180}\right) + \cos\left(\frac{3\pi}{180}\right) + \cos\left(\frac{5\pi}{180}\right) + \ldots + \cos\left(\frac{19\pi}{180}\right) \), we can follow these steps:

### Step 1: Identify the series
The series consists of cosine terms where the angles are in an arithmetic progression (AP). The first term \( a = \frac{\pi}{180} \) and the common difference \( d = \frac{2\pi}{180} = \frac{\pi}{90} \).

### Step 2: Determine the number of terms
The series runs from \( \frac{\pi}{180} \) to \( \frac{19\pi}{180} \). The number of terms \( n \) can be calculated as follows:
- The general term of the AP can be expressed as:
\[ a_n = a + (n-1)d \]
Setting \( a_n = \frac{19\pi}{180} \):
\[ \frac{\pi}{180} + (n-1)\frac{\pi}{90} = \frac{19\pi}{180} \]
Simplifying this gives:
\[ (n-1)\frac{\pi}{90} = \frac{19\pi}{180} - \frac{\pi}{180} = \frac{18\pi}{180} = \frac{\pi}{10} \]
Thus,
\[ n-1 = \frac{\pi/10}{\pi/90} = 9 \implies n = 10 \]

### Step 3: Use the formula for the sum of cosines in AP
The formula for the sum of cosines where the angles are in AP is:
\[ S_n = \frac{\sin\left(\frac{nd}{2}\right) \cos\left(a + \frac{(n-1)d}{2}\right)}{\sin\left(\frac{d}{2}\right)} \]
Substituting the values:
- \( n = 10 \)
- \( a = \frac{\pi}{180} \)
- \( d = \frac{\pi}{90} \)

### Step 4: Calculate each component
1. Calculate \( \frac{nd}{2} \):
\[ \frac{nd}{2} = \frac{10 \cdot \frac{\pi}{90}}{2} = \frac{10\pi}{180} = \frac{\pi}{18} \]

2. Calculate \( a + \frac{(n-1)d}{2} \):
\[ a + \frac{(n-1)d}{2} = \frac{\pi}{180} + \frac{9 \cdot \frac{\pi}{90}}{2} = \frac{\pi}{180} + \frac{9\pi}{180} = \frac{10\pi}{180} = \frac{\pi}{18} \]

3. Calculate \( \frac{d}{2} \):
\[ \frac{d}{2} = \frac{\frac{\pi}{90}}{2} = \frac{\pi}{180} \]

### Step 5: Substitute into the formula
Now substituting these values into the formula:
\[ S_{10} = \frac{\sin\left(\frac{\pi}{18}\right) \cos\left(\frac{\pi}{18}\right)}{\sin\left(\frac{\pi}{180}\right)} \]

### Step 6: Simplify the expression
Using the identity \( \sin(2x) = 2 \sin(x) \cos(x) \):
\[ S_{10} = \frac{1}{2} \cdot \frac{\sin\left(\frac{\pi}{9}\right)}{\sin\left(\frac{\pi}{180}\right)} \]

### Final Result
Thus, the final expression for \( S_{10} \) is:
\[ S_{10} = \frac{1}{2} \cdot \frac{\sin\left(\frac{\pi}{9}\right)}{\sin\left(\frac{\pi}{180}\right)} \]