(a) Show that the equation \[ 5 \cos ^{2} x=3(1+\sin x) \] can be written as \[ 5 \sin ^{2} x+3 \sin x-2=0 \]

Step 1 ：Given the equation \(5 \cos ^{2} x=3(1+\sin x)\)

Step 2 ：We know that \(\cos^2x = 1 - \sin^2x\). We can substitute this into the equation to convert it entirely into terms of sine.

Step 3 ：Substituting \(\cos^2x\) with \(1 - \sin^2x\) in the given equation, we get \(5(1 - \sin^2x) = 3(1+\sin x)\)

Step 4 ：Simplifying the equation, we get \(5 - 5\sin^2x = 3 + 3\sin x\)

Step 5 ：Rearranging the terms, we get \(5\sin^2x + 3\sin x - 2 = 0\)

Step 6 ：\(\boxed{5\sin^2x + 3\sin x - 2 = 0}\) is the final form of the given equation.