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Which expression does (cos 3x)(cos x) − (sin 3x)(sin x) simplify to?cos 4xsin 2xcos 2xsin 4x

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To simplify the expression, we need to remember that the cosine function has the following identity:


\cos\alpha\cos\beta-\sin\alpha\sin\beta=\cos(\alpha+\beta)

Which gives an expression for the cosine of a sum of angles. In this case we have the following original expression:


\cos3x\cos x-\sin3x\sin x

If we compare this expression to the right side of the identity given above, we notice that we have a similar structure but, in this case, instead of alpha we have 3x and instead of beta we have x; for this reason, let:


\begin{gathered} \alpha=3x \\ \beta=x \end{gathered}

Then we have, according to the identity given above:


\cos3x\cos x-\sin3x\sin x=\cos(3x+x)=\cos4x

Therefore, the expression given simplifies to:


\cos4x

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