If `sin 17^@ = x/y`,then the value of `sec 17^@ - sin 73^@` is

Options

A. `(y^2 - x^2)/(xy)`
B. `x^2/(sqrt(y^2 -x^2))`
C. `x^2/(ysqrt(y^2 +x^2))`
D. `x^2/(ysqrt(y^2 -x^2))`

Answer

To solve the problem, we need to find the value of \( \sec 17^\circ - \sin 73^\circ \) given that \( \sin 17^\circ = \frac{x}{y} \).

### Step-by-Step Solution:

1. **Identify the relationship between sine and cosine:**
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Therefore, for \( \theta = 17^\circ \):
\[ \cos^2 17^\circ = 1 - \sin^2 17^\circ \]

2. **Substitute the given value of sine:**
Since \( \sin 17^\circ = \frac{x}{y} \), we can substitute:
\[ \cos^2 17^\circ = 1 - \left(\frac{x}{y}\right)^2 \]
This simplifies to:
\[ \cos^2 17^\circ = 1 - \frac{x^2}{y^2} = \frac{y^2 - x^2}{y^2} \]

3. **Find cosine:**
Taking the square root gives us:
\[ \cos 17^\circ = \sqrt{\frac{y^2 - x^2}{y^2}} = \frac{\sqrt{y^2 - x^2}}{y} \]

4. **Calculate secant:**
The secant function is the reciprocal of cosine:
\[ \sec 17^\circ = \frac{1}{\cos 17^\circ} = \frac{y}{\sqrt{y^2 - x^2}} \]

5. **Evaluate \( \sin 73^\circ \):**
Using the co-function identity:
\[ \sin 73^\circ = \cos(90^\circ - 73^\circ) = \cos 17^\circ \]
We already found that:
\[ \cos 17^\circ = \frac{\sqrt{y^2 - x^2}}{y} \]

6. **Substitute into the expression:**
Now we can substitute \( \sec 17^\circ \) and \( \sin 73^\circ \) into the expression:
\[ \sec 17^\circ - \sin 73^\circ = \frac{y}{\sqrt{y^2 - x^2}} - \frac{\sqrt{y^2 - x^2}}{y} \]

7. **Combine the fractions:**
To combine these fractions, we need a common denominator:
\[ = \frac{y^2 - (y^2 - x^2)}{y \sqrt{y^2 - x^2}} = \frac{x^2}{y \sqrt{y^2 - x^2}} \]

### Final Result:
Thus, the value of \( \sec 17^\circ - \sin 73^\circ \) is:
\[ \frac{x^2}{y \sqrt{y^2 - x^2}} \]