If `(cosectheta+sintheta)/(cosectheta-sintheta)=(5)/(3)` then the value of `sintheta` is equal to

Options

A. a.`(1)/(2)`
B. b.`(1)/(sqrt(2))`
C. c.`(sqrt(3))/(2)`
D. d.1

Answer

To solve the equation \(\frac{\csc \theta + \sin \theta}{\csc \theta - \sin \theta} = \frac{5}{3}\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \frac{\csc \theta + \sin \theta}{\csc \theta - \sin \theta} = \frac{5}{3} \]
Recall that \(\csc \theta = \frac{1}{\sin \theta}\). Substituting this into the equation gives:
\[ \frac{\frac{1}{\sin \theta} + \sin \theta}{\frac{1}{\sin \theta} - \sin \theta} = \frac{5}{3} \]

### Step 2: Simplify the expression
Now, let's simplify the numerator and the denominator:
- The numerator becomes:
\[ \frac{1 + \sin^2 \theta}{\sin \theta} \]
- The denominator becomes:
\[ \frac{1 - \sin^2 \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta} \]

Thus, the equation simplifies to:
\[ \frac{\frac{1 + \sin^2 \theta}{\sin \theta}}{\frac{\cos^2 \theta}{\sin \theta}} = \frac{5}{3} \]
This simplifies to:
\[ \frac{1 + \sin^2 \theta}{\cos^2 \theta} = \frac{5}{3} \]

### Step 3: Cross-multiply
Cross-multiplying gives:
\[ 3(1 + \sin^2 \theta) = 5 \cos^2 \theta \]

### Step 4: Use the Pythagorean identity
Using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\), we can substitute:
\[ 3(1 + \sin^2 \theta) = 5(1 - \sin^2 \theta) \]

### Step 5: Expand and rearrange
Expanding both sides:
\[ 3 + 3 \sin^2 \theta = 5 - 5 \sin^2 \theta \]
Rearranging gives:
\[ 3 \sin^2 \theta + 5 \sin^2 \theta = 5 - 3 \]
\[ 8 \sin^2 \theta = 2 \]

### Step 6: Solve for \(\sin^2 \theta\)
Dividing both sides by 8:
\[ \sin^2 \theta = \frac{2}{8} = \frac{1}{4} \]

### Step 7: Find \(\sin \theta\)
Taking the square root gives:
\[ \sin \theta = \frac{1}{2} \]

### Final Answer
Thus, the value of \(\sin \theta\) is:
\[ \sin \theta = \frac{1}{2} \]