there are 2 important formula in trigonometry:
cos x = 2cos²(x/2) - 1 and cos x = 1 - 2sin²(x/2)
from the first: 1 + cosx = 2cos²(x/2)
and from the second: 1 - cosx = 2sin²(x/2)
if we use these we give: 2sin²(x/2) / sinx = sinx / 2cos²(x/2)
then multiply 2 sides: sin²x = 4 sin²(x/2) cos²(x/2)
in other hand we know: sinx = 2 sin(x/2) cos(x/2)
we prove it!