Use cos (2x) = cos2 (x) - sin2 (x) to establish the following formulas.

a. cos2 (x) = 1 + cos (2x) / 2

b. sin2 (x) = 1 - cos (2x) / 2

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.