Step 1 Given: Given differential equation is csc(y)dx+sec^(2)(x)dy=0 We want to solve this differential equation by saperation of the variables Step 2 Calculation: Given differential equation is csc(y)dx+sec^(2)(x)dy=0 -> sec^(2)(x)dy=-csc(y)dx -> (dy)/(csc(y))=-(dx)/(sec^(2)(x)) This is variable saperable form so integrating both side int (dy)/(csc(y))= -int (dx)/(sec^(2)(x)) Nowcsc (y)=(1)/(sin y),sec(x)=(1)/(cos x) -> int (sin)dy=-int cos^(2)(x)dx -> int sin(y)dy=-int (1+cos(2x))/(2)dx -> -cos(y)=-(1)/(2)(x+(sin(2x))/(2))+c -> cos(y)=(1)/(2)(x+(sin(2x))/(2))+c' ......(where c'=-c) This is required solution Step 3 Answer: The solution of the differential equati on csc(y)dx+sec^(2)(x)dy=0 is equal to cos(y)=(1)/(2)(x+(sin(2x))/(2))+c'