Evaluate `(y^(2))/(b^(2))-(x^(2))/(a^(2))`, whre `x=a tan theta and y= b sec theta`.

Options

A. 0
B. 1
C. `-1`
D. 3

Answer

To evaluate the expression \(\frac{y^2}{b^2} - \frac{x^2}{a^2}\) where \(x = a \tan \theta\) and \(y = b \sec \theta\), we can follow these steps:

### Step 1: Substitute the values of \(x\) and \(y\)
Given:
- \(x = a \tan \theta\)
- \(y = b \sec \theta\)

We can substitute these into the expression:
\[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = \frac{(b \sec \theta)^2}{b^2} - \frac{(a \tan \theta)^2}{a^2} \]

### Step 2: Simplify the fractions
Now simplify each term:
\[ \frac{(b \sec \theta)^2}{b^2} = \frac{b^2 \sec^2 \theta}{b^2} = \sec^2 \theta \]
\[ \frac{(a \tan \theta)^2}{a^2} = \frac{a^2 \tan^2 \theta}{a^2} = \tan^2 \theta \]

### Step 3: Rewrite the expression
Now we can rewrite the expression:
\[ \sec^2 \theta - \tan^2 \theta \]

### Step 4: Use the trigonometric identity
We know from trigonometric identities that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]

### Step 5: Conclusion
Thus, the value of the expression \(\frac{y^2}{b^2} - \frac{x^2}{a^2}\) is:
\[ 1 \]

### Final Answer
The final answer is \(1\).

---