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

Find the value of (cos π₯ + sin π₯)Β² + (cos π₯ β sin π₯)Β².

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### Video Transcript

Find the value of cos π₯ plus sin π₯ squared plus cos π₯ minus sin π₯ squared.

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

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

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

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

Now of course, addition is commutative. It can be performed in any order. So we donβt actually need these brackets. Then we see that two cos π₯ sin π₯ plus negative two cos π₯ sin π₯ is zero. And one plus one is equal to two. And so we found the value of cos π₯ plus sin π₯ squared plus cos π₯ minus sin π₯ squared to be equal to two.