A function \(y=f(x)\) satisfies \(f(x) \sin 2 x+\sin x-\left(1+\cos ^{2} x\right) f^{\prime}(x)=0\) with condition \(f(0…

\( f(x) \cdot \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0 \)
\( \Rightarrow\left(1+\cos ^2 x\right) f^{\prime}(x)-(\sin 2 x) f(x)=\sin x \)
\( \Rightarrow \frac{d}{d x}\left(f(x)\left(1+\cos ^2 x\right)\right)=\sin x \)
\( \Rightarrow f(x)\left(1+\cos ^2 x\right)=-\cos x+c, \text { As } f(0)=0 \Rightarrow c=1 \)
\( \Rightarrow f(x)=\frac{1-\cos ^2 x}{1+\cos ^2 x} \Rightarrow f\left(\frac{\pi}{2}\right)=1\)