\(\sin x dy/dx+ y\cos x=0\)

\(\sin x dy= -y\cos x dx\)

\(dy/y= - \cos x/\sin x dx\)

\(\int dy/y= -\int \cos x/\sin x dx\)

\(\ln |(y)|= -\ln |(\sin x)|+\ln |(C)|\)

\(\ln y= \ln (C/\sin x)\)

\(y\sin x= C\)

Now, We are applying the given Initial Condition is as follow

\(y((7\pi)/6)= -2\)

\(-2*\sin((7\pi)/6)= C\) \(-2* -1= C\) \(\therefore\sin(7\pi/6)= -1\)

C=2

\(y\sin x= 2\)