Let's look at the first equation.
sin2x = sinx
2sinxcosx = sinx (trig identity)
2sinxcosx - sinx = 0 (subtraction)
sinx(2cosx - 1) = 0 (factor)
sinx = 0 or 2cosx -1 = 0 (zero product property)
x=sin-10 or cosx = 1/2
x = 0, π, 2π..., which we can write kπ where k is any integer.
For x = cos-11/2 = π/3, -π/3 ... (to list all solutions: π/3 + 2kπ, k any integer or -π/3+ 2kπ, k any integer)
There are several different ways to represent the set of all solutions, but this gives you the idea.
cos2x + cos2x = -1
cos2x + (cos2x -1) = -1 (trig identity; since cos already appears in the equation choose to use this flavor of cos2x)
2cos2x = 0 (add 1)
cos2x = 0 (division)
cosx = 0 (square root)
x= cos-10 = π/2 and -π/2, but there are infinitely many more solutions. One way to write that is:
x = π/2 + kπ, k any integer