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.