For example by the angle difference identity:

sin(15)=sin(45-30)=sin(45)cos(30)-cos(45)sin(30)

=√2/2*√3/2 - √2/2*1/2

=(√6-√2)/4

Whereas by the half angle identity

sin(15)=sin(30/2)=√((1-cos(30))/2)

=√((1-√3/2)/2)

=√(2-√3)/2

While these are undoubtedly the same, they look very different, especially the second with its radical within a radical. Can you show algebraically that they are indeed equivalent?