So I want to use the power reduction identities that we derived using the cosine double angle identities. Let’s use the power reduction identities to reduce cosine of the fourth power of theta. Well recall that cosine squared theta is 1 plus cosine 2 theta over 2.
So cosine to the fourth power of theta is going to be cosine squared theta squared this formula squared, so let’s do that. 1 plus cosine 2 theta over 2 squared and that will give me 4 in the denominator and in the numerator 1 plus twice the mixed product 2 cosine 2 theta plus cosine squared 2 theta.
Now I’m going to need to use the cosine double angle identity again on this guy. So let me copy this over equals 1 plus 2 cosine 2 theta plus cosine squared 2 theta over 4. Now remember cosine reduction formula is cosine squared equals 1 plus 2 theta over 2. Every time you reduce the power you double the angle. So when I reduce the power here I’m going to get 1 plus cosine 4 theta over 2. All that's divided by 4.
Well let’s get rid of this fraction here and I’ll do that by multiplying the top and bottom by 2, so I’ll get 2 plus 4 cosine 2 theta plus, the 2’s will just cancel in this case, 1 plus cosine 4 theta all over 8. And this simplifies a little bit the 2 in one give me 3, so I’ll write 3/8. 4 cosine 2 theta over 8 is one half cosine 2 theta and finally cosine 4 theta over 8 that’s 1/8 cosine 4 theta. So that’s a reduction identity for cosine to the fourth cosine to the fourth is 3/8 plus a half cosine 2 theta plus 1/8 cosine 4 theta.