If `" cosec"^(2) theta=(625)/(576)`, then what is the value of `[((sin theta - cos theta))/(sin theta + cos theta)]`?

Options

A. `1`
B. `(31)/(17)`
C. `(17)/(31)`
D. `(14)/(25)`

Answer

To solve the problem, we start with the given equation:

\[ \csc^2 \theta = \frac{625}{576} \]

### Step 1: Convert to sine and cosine
We know that:

\[ \csc^2 \theta = 1 + \cot^2 \theta \]

From the definition of cosecant, we also have:

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

Thus,

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

Setting the two expressions for \(\csc^2 \theta\) equal gives us:

\[ \frac{1}{\sin^2 \theta} = \frac{625}{576} \]

### Step 2: Find \(\sin^2 \theta\)
Taking the reciprocal of both sides, we get:

\[ \sin^2 \theta = \frac{576}{625} \]

### Step 3: Find \(\sin \theta\)
Taking the square root of both sides, we find:

\[ \sin \theta = \frac{24}{25} \]

### Step 4: Find \(\cos^2 \theta\)
Using the Pythagorean identity:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

Substituting \(\sin^2 \theta\):

\[ \frac{576}{625} + \cos^2 \theta = 1 \]

This simplifies to:

\[ \cos^2 \theta = 1 - \frac{576}{625} = \frac{625 - 576}{625} = \frac{49}{625} \]

### Step 5: Find \(\cos \theta\)
Taking the square root, we have:

\[ \cos \theta = \frac{7}{25} \]

### Step 6: Calculate \((\sin \theta - \cos \theta)\) and \((\sin \theta + \cos \theta)\)
Now we can find \(\sin \theta - \cos \theta\) and \(\sin \theta + \cos \theta\):

\[ \sin \theta - \cos \theta = \frac{24}{25} - \frac{7}{25} = \frac{24 - 7}{25} = \frac{17}{25} \]

\[ \sin \theta + \cos \theta = \frac{24}{25} + \frac{7}{25} = \frac{24 + 7}{25} = \frac{31}{25} \]

### Step 7: Find the final expression
Now we can find the value of:

\[ \frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} = \frac{\frac{17}{25}}{\frac{31}{25}} = \frac{17}{31} \]

Thus, the final answer is:

\[ \frac{17}{31} \]