Let’s use our two new cosine angle double formulas to derive a power reduction formula. A power reduction formula is a formula that takes you from sine or cosine squared to sine or cosine to the first power.
Now we’ll prove this using a cosine double angle identity cosine 2 theta equals, and let's use the one that involves sine squared 1 minus 2 sine squared theta. Just subtract both from sides cosine 2 theta minus 1 equals negative 2 sine squared theta and then divide both sides by negative 2.
Now we get sine squared theta equals and when I simplify this I can make the denominator positive if I switch the order of the numerator 1 minus cosine 2 theta. And that’s my identity, sine squared theta equals 1 minus cosine 2 theta over 2, very useful in calculus.
Let’s do the power reduction identity for cosine squared. Here I’m going to use cosine 2 theta equals 2 cosine squared theta minus 1. This is the third form, the cosine double angle formula now I just add 1 to both sides cosine 2 theta plus 1 and then divide by 2 and that’s it. Cosine squared theta equals 2 cosine 2 theta plus 1 over 2 and when you look at these formulas notice that the right hand side is almost exactly the same. The only difference is you have a minus for sine squared and a plus for cosine squared.
So just remember that difference cosine squared equals 1 plus cosine 2 theta over 2.