Today's soundtrack is Taylor Swift: 1989, the album that marked her switch from country pop to...pop-pop. It's such a great album - very catchy!
Today, I'm continuing to learn about trigonometric identities.
I was introduced to an important rule today that will really streamline the process of proving identities: We should only multiply terms if doing so will cancel a factor.
We can use Difference of Squares factoring to break down a term so that we can prove an identity. For example, sin²x is equal to 1-cos²x; 1-cos²x is a difference of squares which can be factored as (1-cosx)(1+cosx).
If we find a rational term with a binomial denominator that doesn't have any basic identities, we'll need to use the conjugate: multiply the both numerator and the denominator by a copy of the original denominator, but with a flipped sign in the middle.