If x =p `cosec theta` and y =q `cot theta`, then the value of `(x^2)/(p^2)- (y^2)/(q^2)` is

Options

A. `sin^2 theta`
B. `tan theta`
C. 1
D. 0

Answer

To solve the problem, we start with the given expressions for \( x \) and \( y \):

1. **Given:**
\[ x = p \cdot \csc \theta \]
\[ y = q \cdot \cot \theta \]

2. **We need to find:**
\[ \frac{x^2}{p^2} - \frac{y^2}{q^2} \]

3. **Substituting the values of \( x \) and \( y \):**
\[ \frac{x^2}{p^2} = \frac{(p \cdot \csc \theta)^2}{p^2} = \frac{p^2 \cdot \csc^2 \theta}{p^2} = \csc^2 \theta \]
\[ \frac{y^2}{q^2} = \frac{(q \cdot \cot \theta)^2}{q^2} = \frac{q^2 \cdot \cot^2 \theta}{q^2} = \cot^2 \theta \]

4. **Now, we can rewrite the expression:**
\[ \frac{x^2}{p^2} - \frac{y^2}{q^2} = \csc^2 \theta - \cot^2 \theta \]

5. **Using the trigonometric identity:**
\[ \csc^2 \theta - \cot^2 \theta = 1 \]

6. **Thus, we find:**
\[ \frac{x^2}{p^2} - \frac{y^2}{q^2} = 1 \]

7. **Final answer:**
The value of \( \frac{x^2}{p^2} - \frac{y^2}{q^2} \) is \( 1 \).