Video: Simplifying Trigonometric Expressions Using Double-Angle Identities

Which of the following expressions is equivalent to cos 12𝑥 + sin² 6𝑥? [A] cos² 6𝑥 − sin² 6𝑥 [B] 2 cos² 6𝑥 [C] 2 cos² 6𝑥 + 1 − sin² 6𝑥 [D] 1 − sin² 6𝑥 [E] 2 sin 6𝑥 cos 6𝑥.

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Video Transcript

Which the following expressions is equivalent to cos 12𝑥 plus sin squared six 𝑥? A) cos squared six 𝑥 minus sin squared six 𝑥, B) two cos squared six 𝑥, C) two cos squared six 𝑥 plus one minus sin squared six 𝑥, D) one minus sin squared six 𝑥, or E) two sin six 𝑥 cos six 𝑥.

Well, to be able to actually show which one of the expressions is equivalent to our expression here, what we’re gonna need to use is one of the double angle formulae. And these are a set of formulae that actually, yeah, you need to so to revise and remember. And here’s the one that we’re gonna use today.

So this formula states that the cos two 𝜃 is equal to cos squared 𝜃 minus sin squared 𝜃. To be able to use this in this question, we’re actually gonna say that 𝜃, so the 𝜃 in this formula, is gonna be equal to six 𝑥. So therefore, we can say that the cos two 𝜃 would be the same as the cos 12𝑥. We look back at our original expression. We’ve actually got a cos 12𝑥.

Now because we know it’s the same as cos two 𝜃 in our double angle formula, we can actually express it now like this, which is cos squared six 𝑥 minus sin squared six 𝑥, and then plus our original sin squared six 𝑥. We can actually now simplify this even further cause we’ve got cos squared six 𝑥 then minus sin squared six 𝑥 plus sin squared six 𝑥. So these will actually cancel each other out, which will give us cos squared six 𝑥.

Great! Okay, does this match any of our answers on the left-hand side? But actually it doesn’t yet. So what we’re gonna use, we’re gonna use one of our trigonometric identities. And this identity states that cos squared 𝜃 plus sin squared 𝜃 equals one. And this is probably one of the most common ones that you will use. And then we can actually rearrange it to give us cos squared 𝜃 is equal to one minus sin squared 𝜃.

And great! Now this means this is gonna be something we can use to actually change our cos squared six 𝑥 into a different form, which now how when we see it will give us one minus sin squared six 𝑥, cause we’re gonna use the trig identity to help us get that. So then we can look at our answers on the left-hand side and see that one minus sin squared six 𝑥 is equivalent to cos 12𝑥 plus sin squared six 𝑥.

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