Finding all solutions to a trigonometric equation I was solving a simple trigonometric equation for my brother, going this path: $16\sin^2(2x)\cos^2(2x) = 3$ ${[4\sin(2x)\cos(2x)]}^2 = 3$ Applied the formula for $\sin(2a) = 2\sin(a)\cos(a)$ "backwards" $[2\sin(4x)]^2 = 3$ $\sin(4x) = \frac{\sqrt{3}}{2}$ Then the solutions are easily found: $\frac{\pi}{12} + k\frac{\pi}{2}$ and $\frac{\pi}{6} + k\frac{\pi}{2}$ However the book lists 4 different solutions. I can find them with little effort, but very arbitrarily. How can I know in advance how many solutions there are to such an equation? And to a generic $f(x) = 0$?

You have missed that $[2\sin(4x)]^2=3$ can also mean $\sin(4x)=-\frac{\sqrt3}2$. And being able to see how many solutions there are comes mostly from experience, and it is not infallible.