If ` cos theta = (3)/(5)` , then the value of `(sin theta - tan theta + 1)/( 2 tan^(2) theta)` is

Options

A. `(13)/(15)`
B. `(91)/(160)`
C. `(14)/(15)`
D. `(92)/(160)`

Answer

To solve the problem where \( \cos \theta = \frac{3}{5} \), we need to find the value of \( \frac{\sin \theta - \tan \theta + 1}{2 \tan^2 \theta} \).

### Step 1: Find \( \sin \theta \) and \( \tan \theta \)

Since we know \( \cos \theta = \frac{3}{5} \), we can use the Pythagorean identity to find \( \sin \theta \).

Using the identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
we can substitute \( \cos \theta \):
\[ \sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{9}{25} = 1 \]
Subtract \( \frac{9}{25} \) from both sides:
\[ \sin^2 \theta = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]
Taking the square root gives:
\[ \sin \theta = \frac{4}{5} \quad (\text{since } \theta \text{ is in the first quadrant}) \]

Now, we can find \( \tan \theta \):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]

### Step 2: Substitute \( \sin \theta \) and \( \tan \theta \) into the expression

Now we substitute \( \sin \theta \) and \( \tan \theta \) into the expression:
\[ \frac{\sin \theta - \tan \theta + 1}{2 \tan^2 \theta} = \frac{\frac{4}{5} - \frac{4}{3} + 1}{2 \left(\frac{4}{3}\right)^2} \]

### Step 3: Simplify the numerator

First, we need a common denominator for the numerator:
- The common denominator for \( 5 \) and \( 3 \) is \( 15 \).

Convert each term:
\[ \frac{4}{5} = \frac{12}{15}, \quad \frac{4}{3} = \frac{20}{15}, \quad 1 = \frac{15}{15} \]
Now substitute these into the numerator:
\[ \frac{12}{15} - \frac{20}{15} + \frac{15}{15} = \frac{12 - 20 + 15}{15} = \frac{7}{15} \]

### Step 4: Simplify the denominator

Now calculate \( 2 \tan^2 \theta \):
\[ \tan^2 \theta = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
Thus,
\[ 2 \tan^2 \theta = 2 \cdot \frac{16}{9} = \frac{32}{9} \]

### Step 5: Combine the results

Now we can combine the results:
\[ \frac{\frac{7}{15}}{\frac{32}{9}} = \frac{7}{15} \cdot \frac{9}{32} = \frac{7 \cdot 9}{15 \cdot 32} = \frac{63}{480} \]

### Step 6: Simplify the fraction

Now we simplify \( \frac{63}{480} \):
The greatest common divisor of \( 63 \) and \( 480 \) is \( 3 \):
\[ \frac{63 \div 3}{480 \div 3} = \frac{21}{160} \]

### Final Answer

Thus, the value of \( \frac{\sin \theta - \tan \theta + 1}{2 \tan^2 \theta} \) is:
\[ \frac{21}{160} \]