Video: Using Double Angle Identities to Simplify Trigonometric Expressions

Simplify cos² 52° − sin² 52°.

02:53

Video Transcript

Simplify cos squared 52 minus sin squared 52.

To solve this problem, we’re gonna have to use a compound angle formula. And the formula we’re actually gonna use is this one, which tells us that cos two 𝜃 is equal to cos squared 𝜃 minus sin squared 𝜃, which in turn is equal to cos squared 𝜃 minus one, which is also equal to one minus two sin squared 𝜃.

Now as you can see, this is a lengthy formula that has a number of relationships. So now, we’re gonna have a look at how we can break it down and decide which parts we’re going to use. As we can see from the question, in our expression what we actually do is we have the same angle in both parts. We have 52 degrees. In both terms, it is cos squared 52 degrees minus sin squared 52 degrees.

So if we look back at the formula, we can see that our 𝜃 is equal to 52 degrees. This means that we can actually rewrite our expression as cos squared 𝜃 minus sin squared 𝜃, which is fantastic because what it means is we can now highlight the part of the compound angle formula that we’re going to use. So as you can see here the part I highlighted, this actually represents that: cos squared 𝜃 minus sin squared 𝜃.

But now if you want to fully simplify it, what we need to see is which one of the other relationships we want to choose to actually help us do that. But as already is pointed out, we already know the value of 𝜃. This relationship here would be the best one to choose. So now, we can actually use it back on the left-hand side to write that cos squared 𝜃 minus sin squared 𝜃 is equal to cos two 𝜃.

Right, now, the only thing we need to do now to help us simplify it is actually substitute back in 𝜃 is equal to 52 degrees. So this means we’re gonna get cos squared 52 minus sin squared 52 is equal to the cosine of two multiplied by 52. So therefore, we can say that cos squared 52 minus sin squared 52 is equal to the cosine of 104 degrees fully simplified.

So great! We’ve got to our final answer, but what we’re gonna do now is quickly check it. And in order to do that, we’re actually gonna check it on a calculator. So we can start by putting into our calculator cos squared 52 minus sin squared 52. And this gives us negative 0.24 to two decimal places.

Remembering at this point to make sure that the calculator is in degrees, so it’s gotta little d or deg in the display. So next, we’re now going to find out what the cosine of 104 degrees is. So we type that into our calculators. And as expected, we also get negative 0.24 to two decimal places.

Great! So we can now say that our fully checked answer is that the cos squared 52 minus sin squared 52 is equal to the cosine of 104 degrees.

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