`x=acost,y=asint`. find `dy/dx`

Options

A. `tant`
B. `-tant`
C. `-cott`
D. `cott`

Explanation

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

### Step 1: Differentiate \(x\) and \(y\) with respect to \(t\)

1. Differentiate \(x = a \cos t\):
\[ \frac{dx}{dt} = -a \sin t \]

2. Differentiate \(y = a \sin t\):
\[ \frac{dy}{dt} = a \cos t \]

### Step 2: Find \(\frac{dy}{dx}\)

Using the chain rule, we can find \(\frac{dy}{dx}\) by dividing \(\frac{dy}{dt}\) by \(\frac{dx}{dt}\):
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{a \cos t}{-a \sin t} \]

### Step 3: Simplify the expression

The \(a\) in the numerator and denominator cancels out:
\[ \frac{dy}{dx} = \frac{\cos t}{-\sin t} = -\cot t \]

### Final Answer

Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\cot t \]

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