Video narration
How can you rewrite cos three A using only cos A? Treat it like classwork: each line replaces an expression with an equal one. Split three A into two A plus A, so cos three A becomes cos of two A plus A. Use the cosine addition identity: cosine of A plus B equals cosine A cosine B minus sine A sine B. The next equal line is cos two A cos A minus sin two A sin A. The double-angle identities are known results here, not proved: cos two A is two cosine squared A minus one; sine two A is two sine A cosine A. Substitute: two cosine cubed A minus cosine A minus two sine squared A cosine A. Replace sine squared A with one minus cosine squared A, then collect like terms: four cosine cubed A minus three cosine A. That classwork chain answers it: cos three A equals four cosine cubed A minus three cosine A.