When teaching trigonometry we come across both the difference identities and the half angle identities. Each of these can give what look like very different values for the same thing.

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?

Since both values are positive, we can square them. The first value, squared, produces (8-4√3)/16, and the second (2-√3)/4, which are obviously the same.

Posted by Old Original Oskar! on 2007-06-25 09:18:49