Expansion and Reduction of Trigonometric Expressions

In this exercise, we will expand and simplify three trigonometric expressions. Let's start with the first one.

The first expression is the sum of two squared binomials: cosine x plus sine x quantity squared, plus cosine x minus sine x quantity squared.

We use the algebraic identity for a plus b squared and a minus b squared to expand both parts.

Notice that the middle terms, two cosine x sine x and negative two cosine x sine x, cancel each other out.

Using the Pythagorean identity, cosine squared plus sine squared equals one. So we have one plus one.

The result for the first part is simply two.

Now let's look at the second expression. It looks like a difference of squares if we group the sine and cosine terms together.

By treating sine x plus cosine x as a single term, we can apply the identity for a plus b times a minus b.