Given a and b are unit vectors

So, |a| = 1, |b| = 1

Again given the angle between the unit vectors, a and b is θ 

Now, Using the law of cosines on the triangle formed by vector a, b, we get

      |a - b|² =  |a|² + |b|² - 2cosθ

=> |a - b|² =  1 + 1 - 2cosθ

=> |a - b|² =  2 - 2cosθ

Since, cosθ = 1 - 2 sin² (θ/2)

Now,  |a - b|² = 2 - 2{1 - 2sin² (θ/2)}

=> |a - b|² = 2 - 2 + 4sin² (θ/2)

=> |a - b|² = 4sin² (θ/2)

=> |a - b| = √{4sin² (θ/2)}

=> |a - b| = 2sin(θ/2)}

=> sin(θ/2) = |a - b|/2