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 \(x\) with respect to \(t\)
Given:
\[ x = 2\cos t - \cos 2t \]
To find \(\frac{dx}{dt}\), we differentiate \(x\):
\[ \frac{dx}{dt} = \frac{d}{dt}(2\cos t) - \frac{d}{dt}(\cos 2t) \]
Using the derivative of cosine:
\[ \frac{dx}{dt} = -2\sin t + 2\sin 2t \]

### Step 2: Differentiate \(y\) with respect to \(t\)
Given:
\[ y = 2\sin t - \sin 2t \]
To find \(\frac{dy}{dt}\), we differentiate \(y\):
\[ \frac{dy}{dt} = \frac{d}{dt}(2\sin t) - \frac{d}{dt}(\sin 2t) \]
Using the derivative of sine:
\[ \frac{dy}{dt} = 2\cos t - 2\cos 2t \]

### Step 3: Evaluate \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) at \(t = \frac{\pi}{4}\)
Now we will substitute \(t = \frac{\pi}{4}\) into both derivatives.

**For \(\frac{dx}{dt}\)**:
\[ \frac{dx}{dt} = -2\sin\left(\frac{\pi}{4}\right) + 2\sin\left(2 \cdot \frac{\pi}{4}\right) \]
\[ = -2\cdot\frac{1}{\sqrt{2}} + 2\cdot\sin\left(\frac{\pi}{2}\right) \]
\[ = -\frac{2}{\sqrt{2}} + 2\cdot 1 = -\sqrt{2} + 2 \]

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

### Step 4: Calculate \(\frac{dy}{dx}\)
Now we can find \(\frac{dy}{dx}\) using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the values we found:
\[ \frac{dy}{dx} = \frac{\sqrt{2}}{2 - \sqrt{2}} \]

### Step 5: Simplify \(\frac{dy}{dx}\)
To simplify:
\[ \frac{dy}{dx} = \frac{\sqrt{2}}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}} = \frac{\sqrt{2}(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} \]
The denominator simplifies to:
\[ (2 - \sqrt{2})(2 + \sqrt{2}) = 4 - 2 = 2 \]
Thus:
\[ \frac{dy}{dx} = \frac{\sqrt{2}(2 + \sqrt{2})}{2} = \frac{2\sqrt{2} + 2}{2} = \sqrt{2} + 1 \]

### Final Answer
\[ \frac{dy}{dx} = \sqrt{2} + 1 \]