Question

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

Answer 1

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

b. sin2(x) = 1 − cos(2x) / 2

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.