Other Forms of the Cosine Double-Angle Formula - Concept
The cosine double angle formula is cos(2theta)=cos2(theta) - sin2(theta). Combining this formula with the Pythagorean Identity, cos2(theta) + sin2(theta)=1, two other forms appear: cos(2theta)=2cos2(theta)-1 and cos(2theta)=1-2sin2(theta). These can be used to find the power-reduction formulas, which reduce a second degree or higher trig function to a first degree. These formulas are very useful in Calculus.
I want to talk about other forms of the cosine double angle identities. First, let's recall the Pythagorean identity and the two other forms of it. Cosine squared plus sine squared equals 1 can also be written cosine squared theta equals 1 minus sine squared theta or sine squared theta equals 1 minus cosine squared theta. Now the original cosine double angle formula is this, cosine of 2 theta equals cosine squared theta minus sine squared theta, but I can use my Pythagorean identities to rewrite this, so another form would be cosine oops cosine 2 theta equals alpha cosine theta I'll replace it with 1 minus sine squared theta minus sine theta sine squared theta and that's 1 minus 2 sine squared theta so that's the second form cosine 2 theta equals 1 minus 2 sine squared theta but we can also do cosine of 2 theta equals and starting from here I can replace sine squared with 1 minus cosine so I get cosine squared theta minus 1 minus sine squared theta and the minus distribute I get minus 1 I'm sorry this should be cosine, here we go, our minus 1 distributes we get minus 1 and minus minus plus cosine squared theta so cosine squared theta minus 1 plus cosine squared theta is 2 cosine squared theta minus 1 they're very similar. Cosine 2 theta is 1 minus 2 sine squared theta, cosine 2 theta equals 2 cosine squared minus 1.
In order to remember which is which remember the original cosine double angle formula cosine is the one that's positive sine is the one that it's negative so in the other forms sine is still negative and cosine is still positive.