Prove 2 trigonometric identities.?

Question

a. abs(cos(x/2)) = sqrt((1+cosx)/2)

b. abs(sin(x/2)) = sqrt((1-cosx)/2)

Answer 1

we know, cos^2 x/2 - sin^2 x/2 = cos x .......... (1)

& we also know, sin^2 x/2 + cos^2 x/2 = 1

=> sin^2 x/2 = 1 - cos^2 x/2

now putting the value of sin^2 x/2 in (1) we have,

cos^2 x/2 - (1 - cos^2 x/2) = cos x

=> cos^2 x/2 - 1 + cos^2 x/2 = cos x

=> 2cos^2 x/2 = 1 + cos x

=> cos^2 x/2 = (1 + cos x)/2

=> cos x/2 = +/- sqrt{(1 + cos x)/2}

so we can say that (a). abs(cos (x/2)) = sqrt{(1 + cos x)/2}

again, putting, cos^2 x/2 = 1 - sin^2 x/2 in (1) we have,

1 - sin^2 x/2 - sin^2 x/2 = cos x

=> 1 - 2sin^2 x/2 = cos x

=> 2sin^2 x/2 = 1 - cos x

=> sin^2 x/2 = (1 - cos x)/2

=> sin x/2 = +/- sqrt{(1 - cos x)/2}

similarly we can say that,

abs(sin (x/2)) = sqrt{(1 - cos x)/2}

Hope this helps u :)