`x =a tan theta ,y =b sec theta` find `dy/dx`

Answer

To find \( \frac{dy}{dx} \) given the equations \( x = a \tan \theta \) and \( y = b \sec \theta \), we can follow these steps:

### Step 1: Differentiate \( x \) with respect to \( \theta \)
We start with the equation for \( x \):
\[ x = a \tan \theta \]
Differentiating both sides with respect to \( \theta \):
\[ \frac{dx}{d\theta} = a \sec^2 \theta \]
Let’s label this as Equation (1).

### Step 2: Differentiate \( y \) with respect to \( \theta \)
Now we differentiate \( y \):
\[ y = b \sec \theta \]
Differentiating both sides with respect to \( \theta \):
\[ \frac{dy}{d\theta} = b \sec \theta \tan \theta \]
Let’s label this as Equation (2).

### Step 3: Find \( \frac{dy}{dx} \)
To find \( \frac{dy}{dx} \), we use the chain rule:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]
Substituting the expressions from Equations (1) and (2):
\[ \frac{dy}{dx} = \frac{b \sec \theta \tan \theta}{a \sec^2 \theta} \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ \frac{dy}{dx} = \frac{b \tan \theta}{a \sec \theta} \]
We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Thus, we can rewrite:
\[ \frac{dy}{dx} = \frac{b \cdot \frac{\sin \theta}{\cos \theta}}{a \cdot \frac{1}{\cos \theta}} = \frac{b \sin \theta}{a} \]

### Final Result
Thus, the final result for \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{b}{a} \sin \theta \]