If `x = sint cos2t` and `y = cost sin2t` , then at `t = pi/4`, the value of `dy/dx` is equal to:

Options

A. -2
B. 2
C. `1/2`
D. `-1/2`

Answer

To find the value of \(\frac{dy}{dx}\) at \(t = \frac{\pi}{4}\) for the given parametric equations \(x = \sin t \cos 2t\) and \(y = \cos t \sin 2t\), we will follow these steps:

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

Given:
\[ y = \cos t \sin 2t \]

Using the product rule:
\[ \frac{dy}{dt} = \frac{d}{dt}(\cos t) \cdot \sin 2t + \cos t \cdot \frac{d}{dt}(\sin 2t) \]

Calculating each derivative:
- \(\frac{d}{dt}(\cos t) = -\sin t\)
- \(\frac{d}{dt}(\sin 2t) = 2\cos 2t\)

Thus,
\[ \frac{dy}{dt} = -\sin t \sin 2t + \cos t \cdot 2\cos 2t \]

### Step 2: Differentiate \(x\) with respect to \(t\)

Given:
\[ x = \sin t \cos 2t \]

Using the product rule:
\[ \frac{dx}{dt} = \frac{d}{dt}(\sin t) \cdot \cos 2t + \sin t \cdot \frac{d}{dt}(\cos 2t) \]

Calculating each derivative:
- \(\frac{d}{dt}(\sin t) = \cos t\)
- \(\frac{d}{dt}(\cos 2t) = -2\sin 2t\)

Thus,
\[ \frac{dx}{dt} = \cos t \cos 2t - 2\sin t \sin 2t \]

### Step 3: Evaluate \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) at \(t = \frac{\pi}{4}\)

Substituting \(t = \frac{\pi}{4}\):
- \(\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}\)
- \(\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\)
- \(\sin \frac{\pi}{2} = 1\)
- \(\cos \frac{\pi}{2} = 0\)

Calculating \(\frac{dy}{dt}\):
\[ \frac{dy}{dt} = -\sin\left(\frac{\pi}{4}\right) \sin\left(\frac{\pi}{2}\right) + \cos\left(\frac{\pi}{4}\right) \cdot 2\cos\left(\frac{\pi}{2}\right) \]
\[ = -\frac{1}{\sqrt{2}} \cdot 1 + \frac{1}{\sqrt{2}} \cdot 2 \cdot 0 = -\frac{1}{\sqrt{2}} \]

Calculating \(\frac{dx}{dt}\):
\[ \frac{dx}{dt} = \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{2}\right) - 2\sin\left(\frac{\pi}{4}\right) \sin\left(\frac{\pi}{2}\right) \]
\[ = \frac{1}{\sqrt{2}} \cdot 0 - 2 \cdot \frac{1}{\sqrt{2}} \cdot 1 = -\frac{2}{\sqrt{2}} = -\sqrt{2} \]

### Step 4: Calculate \(\frac{dy}{dx}\)

Using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the values:
\[ \frac{dy}{dx} = \frac{-\frac{1}{\sqrt{2}}}{-\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2} \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) at \(t = \frac{\pi}{4}\) is:
\[ \frac{1}{2} \]

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