If `(cosectheta+sintheta)/(cosectheta-sintheta)=(5)/(3)` and `0^(@)ltthetalt90^(@)` then what is the value of `tan theta` ?

Options

A. `(1)/(2)`
B. `(1)/(sqrt(3))`
C. `sqrt(3)`
D. `(sqrt(3))/(2)`

Answer

To solve the equation \(\frac{\csc \theta + \sin \theta}{\csc \theta - \sin \theta} = \frac{5}{3}\) and find the value of \(\tan \theta\), we can follow these steps:

### Step 1: Rewrite \(\csc \theta\)
We know that \(\csc \theta = \frac{1}{\sin \theta}\). Therefore, we can substitute this into the equation:

\[ \frac{\frac{1}{\sin \theta} + \sin \theta}{\frac{1}{\sin \theta} - \sin \theta} = \frac{5}{3} \]

### Step 2: Simplify the expression
Now, we can simplify the numerator and the denominator:

**Numerator:**
\[ \frac{1 + \sin^2 \theta}{\sin \theta} \]

**Denominator:**
\[ \frac{1 - \sin^2 \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta} \]

Putting it all together, we have:

\[ \frac{\frac{1 + \sin^2 \theta}{\sin \theta}}{\frac{\cos^2 \theta}{\sin \theta}} = \frac{1 + \sin^2 \theta}{\cos^2 \theta} \]

### Step 3: Set the equation
Now we set the simplified expression equal to \(\frac{5}{3}\):

\[ \frac{1 + \sin^2 \theta}{\cos^2 \theta} = \frac{5}{3} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:

\[ 3(1 + \sin^2 \theta) = 5 \cos^2 \theta \]

### Step 5: Use the Pythagorean identity
Recall that \(\cos^2 \theta = 1 - \sin^2 \theta\). Substitute this into the equation:

\[ 3(1 + \sin^2 \theta) = 5(1 - \sin^2 \theta) \]

### Step 6: Expand and rearrange
Expanding both sides gives:

\[ 3 + 3\sin^2 \theta = 5 - 5\sin^2 \theta \]

Now, rearranging the equation:

\[ 3\sin^2 \theta + 5\sin^2 \theta = 5 - 3 \]

\[ 8\sin^2 \theta = 2 \]

### Step 7: Solve for \(\sin^2 \theta\)
Dividing both sides by 8:

\[ \sin^2 \theta = \frac{2}{8} = \frac{1}{4} \]

### Step 8: Find \(\sin \theta\)
Taking the square root gives:

\[ \sin \theta = \frac{1}{2} \]

### Step 9: Find \(\cos \theta\)
Using the Pythagorean identity:

\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{1}{4} = \frac{3}{4} \]

Thus,

\[ \cos \theta = \frac{\sqrt{3}}{2} \]

### Step 10: Calculate \(\tan \theta\)
Finally, we can find \(\tan \theta\):

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

### Final Answer
Therefore, the value of \(\tan \theta\) is:

\[ \tan \theta = \frac{1}{\sqrt{3}} \]

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