2 cos2() − sin() = 1

The given equation is: 2cos^2(x) - sin(x) = 1

Let's simplify the equation:

2cos^2(x) - sin(x) = 1
1 - sin^2(x) - sin(x) = 1
1 - sin^2(x) - sin(x) - 1 = 0
-sin^2(x) - sin(x) = 0

Now, let's factor out sin(x) from the equation:

-sin(x)(sin(x) + 1) = 0

Setting each factor equal to zero:

sin(x) = 0 or sin(x) + 1 = 0

If sin(x) = 0, then x can be any integer multiple of π.

If sin(x) + 1 = 0, then sin(x) = -1. This occurs when x = -π/2 + 2πn, where n is an integer.

Therefore, the solutions to the equation 2cos^2(x) - sin(x) = 1 are x = πn and x = -π/2 + 2πn, where n is an integer.