\(\text { If } y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right) \text { if } x(1)=\frac{\pi}{2} \text { …

\(\begin{aligned}& y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right) \\& 1=\left(\frac{x}{y}-\frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right) \\& \frac{x}{y}=v \\& \frac{d x}{d y}=v+y \frac{d v}{d y} \\& 1=\left(v-\left(v+y \frac{d v}{d y}\right)\right) \sin v \\& 1=v-v-y \frac{d v}{d y} \cdot \sin v \\& 1=-y \frac{d v}{d y} \cdot \sin v \\& \frac{d y}{y}=-\sin v d v \\& \ln y=\cos v+c \\& \ln y=\cos \frac{x}{y}+c \\& 0=0+c \\& c=0 \\& \ln y=\cos \frac{x}{y} \\& \ln 2=\cos \left(\frac{x}{2}\right) \\\end{aligned}\)

\(\begin{aligned}& \cos x=2 \cos ^2 \frac{x}{2}-1 \\& =2 \ln ^2 2-1\end{aligned}\)