In Exercises 69–74, rewrite each expression as a simplified expre...

Question

In Exercises 69–74, rewrite each expression as a simplified expression containing one term.

cos (α + β) cos β + sin (α + β) sin β

Accepted Answer

Hello. Today we're going to be simplifying the given expression. What we are given is cosine of three X plus two Y multiplied by cosine of three X plus sine of three X plus two Y multiplied by sine of three X. Now one way we can simplify this quantity is by using the difference property of cosine the difference property of cosine states that if you have cosine A minus B, you can rewrite this quantity as cosine of A multiplied by cosine of B plus sine of A multiplied by S sign of B. In this case, we're going to be using this property to go from right to left. But we need to determine our A and B values while A trick into doing this is to note that the first cosine value and the first sign value both have the same angle and the second cosine value and second sine value have the same angle. Now, if we take a look at our original quantity, the first code sine value and the first sine value have the same angle of three X plus two Y. So we're going to allow A to equal to the quantity three X plus two Y. And if we take a look at the second sign and cosine values, they both have an angle of three X. So we're going to allow our B to equal to three X. So now that we're gonna use this identity to simplify the product. And with the given A and B values, we can rewrite our original expression as cosine of A which is three X plus two Y minus B, which is three X. Now, we need to simplify the insight argument for cosine three X minus three X is going to cancel the quantities out. And what we are left with is cosine of two Y. And this is going to be the simplified form of the original expression. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 69–74, rewrite each expression as a simplified expression containing one term. cos (α + β) cos β + sin (α + β) sin β

Key Concepts

Angle Addition Formulas

Trigonometric Identities

Simplification of Trigonometric Expressions

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