If `x=2cost-cos2t`,`y=2sint-sin2t`,then at `t=(pi)/(4)`,`(dy)/(dx)=`

Options

A. `sqrt2+1`
B. `sqrt(2+1)`
C. `(sqrt(2+1))/(2)`
D. none of these

Explanation

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

### Step 1: Differentiate \(y\) with respect to \(t\)
We start by differentiating \(y\) with respect to \(t\):
\[ y = 2\sin t - \sin 2t \]
Using the chain rule, we differentiate:
\[ \frac{dy}{dt} = 2\cos t - \frac{d}{dt}(\sin 2t) \]
Using the derivative of \(\sin 2t\), which is \(2\cos 2t\):
\[ \frac{dy}{dt} = 2\cos t - 2\cos 2t \]

### Step 2: Differentiate \(x\) with respect to \(t\)
Next, we differentiate \(x\) with respect to \(t\):
\[ x = 2\cos t - \cos 2t \]
Differentiating gives:
\[ \frac{dx}{dt} = -2\sin t + \frac{d}{dt}(\cos 2t) \]
Using the derivative of \(\cos 2t\), which is \(-2\sin 2t\):
\[ \frac{dx}{dt} = -2\sin t + 2\sin 2t \]

### Step 3: Find \(\frac{dy}{dx}\)
Now we can find \(\frac{dy}{dx}\) using the formula:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{2\cos t - 2\cos 2t}{-2\sin t + 2\sin 2t} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{2(\cos t - \cos 2t)}{2(-\sin t + \sin 2t)} = \frac{\cos t - \cos 2t}{-\sin t + \sin 2t} \]

### Step 4: Evaluate \(\frac{dy}{dx}\) at \(t = \frac{\pi}{4}\)
Now we substitute \(t = \frac{\pi}{4}\):
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}, \quad \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Also, we need \(\cos\left(\frac{\pi}{2}\right) = 0\) and \(\sin\left(\frac{\pi}{2}\right) = 1\):
\[ \frac{dy}{dx} = \frac{\frac{1}{\sqrt{2}} - 0}{-\frac{1}{\sqrt{2}} + 1} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{\frac{1}{\sqrt{2}}}{-\frac{1}{\sqrt{2}} + 1} = \frac{\frac{1}{\sqrt{2}}}{1 - \frac{1}{\sqrt{2}}} \]
To simplify further:
\[ = \frac{1}{\sqrt{2}(1 - \frac{1}{\sqrt{2}})} = \frac{1}{\sqrt{2} \cdot \frac{\sqrt{2} - 1}{\sqrt{2}}} = \frac{1}{\sqrt{2} - 1} \]
Now, rationalizing the denominator:
\[ = \frac{1(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} = \frac{\sqrt{2} + 1}{2 - 1} = \sqrt{2} + 1 \]

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