Solve the initial value problem: \( \frac{d y}{d x}=\frac{-0.5}{\cos (y)}, \quad y(0)=\frac{\pi}{4} \) \( y(x)= \)

\[ \textbf{Step 1: Separate variables.} \] Given the differential equation \[ \frac{dy}{dx} = \frac{-0.5}{\cos(y)}, \] we can rewrite it as \[ \cos(y) \, dy = -0.5 \, dx. \] \[ \textbf{Step 2: Integrate both sides.} \] Integrate the left side with respect to \( y \) and the right side with respect to \( x \): \[ \int \cos(y) \, dy = \int -0.5 \, dx. \] The integrals become: \[ \sin(y) = -0.5 x + C, \] where \( C \) is the constant of integration. \[ \textbf{Step 3: Apply the initial condition.} \] The initial condition is \[ y(0) = \frac{\pi}{4}. \] Substitute \( x=0 \) and \( y=\frac{\pi}{4} \) into the integrated equation: \[ \sin\left(\frac{\pi}{4}\right) = -0.5 \cdot 0 + C. \] Since \[ \sin\left(\frac{\pi}{4}\right)= \frac{\sqrt{2}}{2}, \] we get \[ C = \frac{\sqrt{2}}{2}. \] \[ \textbf{Step 4: Write the solution.} \] Substituting \( C \) back into the integrated equation, we obtain: \[ \sin(y) = -0.5 x + \frac{\sqrt{2}}{2}. \] To solve for \( y \), take the arcsine of both sides: \[ y = \arcsin\left(-0.5 x + \frac{\sqrt{2}}{2}\right). \] \[ \boxed{y(x) = \arcsin\left(-0.5 x + \frac{\sqrt{2}}{2}\right)} \]