Explanation:

To find the angle the tangent makes with the x-axis, we start by calculating the derivatives dx/dt and dy/dt, and then find dy/dx by dividing these derivatives.

- Given:

- \(x = \cos(3 - 2\cos^2 t)\)

- \(y = \sin(3 - 2\sin^2 t)\)

- Derivatives:

- \(\frac{dx}{dt} = -\sin(3 - 2\cos^2 t) \cdot (-4\cos t \cdot \sin t)\)

- \(\frac{dy}{dt} = \cos(3 - 2\sin^2 t) \cdot (-4\sin t \cdot \cos t)\)

- At \( t = \frac{\pi}{4} \):

- \(\cos t = \sin t = \frac{\sqrt{2}}{2}\)

- \(\frac{dx}{dt} = 2\sqrt{2} \sin(1)\)

- \(\frac{dy}{dt} = -2\sqrt{2} \cos(1)\)

- Tangent slope \(\left(\frac{dy}{dx}\right) = \frac{dy/dt}{dx/dt} = -\frac{\cos(1)}{\sin(1)} = -\cot(1)\)

- Angle with x-axis is \(\tan^{-1}\left(-\frac{\cos(1)}{\sin(1)}\right) = \pi - 1\)

- Compare this with options:

- Option 1: \(0\)

- Option 2: \(\frac{\pi}{4}\)

- Option 3: \(\frac{\pi}{6}\)

- Option 4: \(\pi - 1 \approx \frac{\pi}{3}\)

- Correct Answer: Option 4 - \(\pi/3\)