Answer:
a. cos2(x) = 1 + cos(2x) / 2
b. sin2(x) = 1 − cos(2x) / 2
Step-by-step explanation:
From cos(2x) = cos2(x) − sin2(x)
a. cos2(x) = cos(2x) + sin2(x)
but sin2(x) = 1 - cos2(x)
Therefore,
cos2(x) = cos(2x) + 1 - cos2(x)
cos2(x) + cos2(x) = cos(2x) + 1
2 cos2(x) = cos(2x) + 1
cos2(x) = (cos(2x) + 1)/2
Hence cos2(x) = 1 + cos(2x) / 2
b. sin2(x) = 1 − cos(2x) / 2
cos2(x) = 1 - sin2(x)
Therefore,
sin2(x) = cos2(x) - cos(2x)
sin2(x) = 1 - sin2(x) - cos(2x)
2sin2(x) = 1 - cos(2x)
sin2(x) = (1 - cos(2x))/2
Hence the proof.
