To solve for x in the equation 1-2sin(x) = (cos(x/2) - sin(x/2))^2, we need to simplify the equation first.
1-2sin(x) = (cos(x/2) - sin(x/2))^21 - 2sin(x) = cos^2(x/2) - 2sin(x/2)cos(x/2) + sin^2(x/2)1 - 2sin(x) = cos^2(x/2) - sin^2(x/2) - 2sin(x/2)cos(x/2)
Using the trigonometric identity cos^2(x/2) - sin^2(x/2) = cos(x), we get:
1 - 2sin(x) = cos(x) - 2sin(x/2)cos(x/2)
Now, we can use the half-angle identities to simplify further:
1 - 2sin(x) = cos(x) - sin(x)1 - 2sin(x) = cos(x) - 2sin(x/2)cos(x/2)
Now, we can substitute sin(x) = 1 - cos(x) into the equation:
1 - 2(1 - cos(x)) = cos(x) - 2sin(x/2)cos(x/2)1 - 2 + 2cos(x) = cos(x) - 2sin(x/2)cos(x/2)2cos(x) - 1 = cos(x) - 2sin(x/2)cos(x/2)
Now, we can use the double angle identity cos(2x) = 2cos^2(x) - 1 to simplify further:
2cos(x) - 1 = cos(x) - 2(2cos(x/2)sin(x/2))2cos(x) - 1 = cos(x) - 4sin(x/2)cos(x/2)2cos(x) - 1 = cos(x) - 2sin(x)2cos(x) - cos(x) - 1 = 2sin(x)cos(x) = 2sin(x) + 1
Now, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to solve for x:
cos(x) = 2sin(x) + 11 - sin^2(x) = 2sin(x) + 1sin^2(x) + 2sin(x) = 0sin(x)(sin(x) + 2)sin(x) = 0 or sin(x) = -2 (Not possible)
Therefore, sin(x) = 0x = sin^(-1)(0)x = 0, π
To solve for x in the equation 1-2sin(x) = (cos(x/2) - sin(x/2))^2, we need to simplify the equation first.
1-2sin(x) = (cos(x/2) - sin(x/2))^2
1 - 2sin(x) = cos^2(x/2) - 2sin(x/2)cos(x/2) + sin^2(x/2)
1 - 2sin(x) = cos^2(x/2) - sin^2(x/2) - 2sin(x/2)cos(x/2)
Using the trigonometric identity cos^2(x/2) - sin^2(x/2) = cos(x), we get:
1 - 2sin(x) = cos(x) - 2sin(x/2)cos(x/2)
Now, we can use the half-angle identities to simplify further:
1 - 2sin(x) = cos(x) - sin(x)
1 - 2sin(x) = cos(x) - 2sin(x/2)cos(x/2)
Now, we can substitute sin(x) = 1 - cos(x) into the equation:
1 - 2(1 - cos(x)) = cos(x) - 2sin(x/2)cos(x/2)
1 - 2 + 2cos(x) = cos(x) - 2sin(x/2)cos(x/2)
2cos(x) - 1 = cos(x) - 2sin(x/2)cos(x/2)
Now, we can use the double angle identity cos(2x) = 2cos^2(x) - 1 to simplify further:
2cos(x) - 1 = cos(x) - 2(2cos(x/2)sin(x/2))
2cos(x) - 1 = cos(x) - 4sin(x/2)cos(x/2)
2cos(x) - 1 = cos(x) - 2sin(x)
2cos(x) - cos(x) - 1 = 2sin(x)
cos(x) = 2sin(x) + 1
Now, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to solve for x:
cos(x) = 2sin(x) + 1
1 - sin^2(x) = 2sin(x) + 1
sin^2(x) + 2sin(x) = 0
sin(x)(sin(x) + 2)
sin(x) = 0 or sin(x) = -2 (Not possible)
Therefore, sin(x) = 0
x = sin^(-1)(0)
x = 0, π