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Other Forms of the Cosine Double-Angle Formula - Problem 1
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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.
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