Question Video: Applying the Pythagorean Identities to Evaluate Some Expressions Mathematics

Video Transcript

Find the value of cos 𝑥 plus sin 𝑥 squared plus cos 𝑥 minus sin 𝑥 squared.

Now at first glance, it might not be instantly obvious what we need to do here. So we’re going to begin by simply distributing each pair of parentheses. So we’ll write cos 𝑥 plus sin 𝑥 squared as cos 𝑥 plus sin 𝑥 times cos 𝑥 plus sin 𝑥.

We begin by multiplying the first term in each expression. Cos 𝑥 times cos 𝑥 is cos squared 𝑥. We then multiply the outer terms. And we get cos 𝑥 sin 𝑥. We multiply the inner terms to give us another cos 𝑥 sin 𝑥. And finally, we multiply the last two terms. And we get sin squared 𝑥.

And it follows that cos 𝑥 sin 𝑥 plus cos 𝑥 sin 𝑥 is equal to two cos 𝑥 sin 𝑥. But of course, we know that the Pythagorean identity tells us that cos squared 𝑥 plus sin squared 𝑥 is equal to one. So we find that cos 𝑥 plus sin 𝑥 all squared is equal to two cos 𝑥 sin 𝑥 plus one.

Let’s repeat this process for cos 𝑥 minus sin 𝑥 squared, first writing it as cos 𝑥 minus sin 𝑥 times cos 𝑥 minus sin 𝑥. Distributing these parentheses, and we get cos squared 𝑥 minus cos 𝑥 sin 𝑥 minus cos 𝑥 sin 𝑥 plus sin squared 𝑥. We gather like terms, and we get negative two cos 𝑥 sin 𝑥. And once again, we use the Pythagorean identity that cos squared 𝑥 plus sin squared 𝑥 is equal to one. We can use these two expressions, and we can write cos 𝑥 plus sin 𝑥 squared plus cos 𝑥 minus sin 𝑥 squared is equal to two cos 𝑥 sin 𝑥 plus one plus negative two cos 𝑥 sin 𝑥 plus one.

Now of course, addition is commutative. It can be performed in any order. So we don’t actually need these brackets. Then we see that two cos 𝑥 sin 𝑥 plus negative two cos 𝑥 sin 𝑥 is zero. And one plus one is equal to two. And so we found the value of cos 𝑥 plus sin 𝑥 squared plus cos 𝑥 minus sin 𝑥 squared to be equal to two.