x=a cos t, y=b sin t find `dy/dx`

Answer

To find \( \frac{dy}{dx} \) given the parametric equations \( x = a \cos t \) and \( y = b \sin t \), we can follow these steps:

### Step 1: Differentiate \( x \) with respect to \( t \)
Given:
\[ x = a \cos t \]
Differentiating both sides with respect to \( t \):
\[ \frac{dx}{dt} = -a \sin t \]
This is our first equation.

### Step 2: Differentiate \( y \) with respect to \( t \)
Given:
\[ y = b \sin t \]
Differentiating both sides with respect to \( t \):
\[ \frac{dy}{dt} = b \cos t \]
This is our second equation.

### Step 3: Use the chain rule to find \( \frac{dy}{dx} \)
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the values from our previous steps:
\[ \frac{dy}{dx} = \frac{b \cos t}{-a \sin t} \]

### Step 4: Simplify the expression
We can simplify this expression:
\[ \frac{dy}{dx} = -\frac{b}{a} \cdot \frac{\cos t}{\sin t} \]
This can be rewritten using the tangent function:
\[ \frac{dy}{dx} = -\frac{b}{a} \cdot \cot t \]
Or, alternatively:
\[ \frac{dy}{dx} = -\frac{a}{b} \tan t \]

### Final Answer
Thus, the value of \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{a}{b} \tan t \]