Find `dy/dx`, if x and y are connected parametrically by the equations, given below without eliminating the parameter:
`y=12(1-cost),x=10(t-sint),-pi/2lttltpi/2`.

Answer

To find \(\frac{dy}{dx}\) for the given parametric equations \(y = 12(1 - \cos t)\) and \(x = 10t - \sin t\), we will use the parametric form of differentiation.

### Step-by-step Solution:

1. **Identify the Parametric Equations**:
\[ y = 12(1 - \cos t) \]
\[ x = 10t - \sin t \]

2. **Differentiate \(y\) with respect to \(t\)**:
To find \(\frac{dy}{dt}\), we differentiate \(y\):
\[ \frac{dy}{dt} = \frac{d}{dt}[12(1 - \cos t)] = 12 \cdot \frac{d}{dt}(1 - \cos t) \]
The derivative of \(1\) is \(0\) and the derivative of \(-\cos t\) is \(\sin t\):
\[ \frac{dy}{dt} = 12 \sin t \]

3. **Differentiate \(x\) with respect to \(t\)**:
Now, we find \(\frac{dx}{dt}\):
\[ \frac{dx}{dt} = \frac{d}{dt}[10t - \sin t] = 10 - \cos t \]

4. **Use the Parametric Derivative Formula**:
The formula for \(\frac{dy}{dx}\) in parametric form is:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{12 \sin t}{10 - \cos t} \]

5. **Final Result**:
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{12 \sin t}{10 - \cos t} \]