To simplify the expression, we need to remember that the cosine function has the following identity:
[tex]\cos\alpha\cos\beta-\sin\alpha\sin\beta=\cos(\alpha+\beta)[/tex]Which gives an expression for the cosine of a sum of angles. In this case we have the following original expression:
[tex]\cos3x\cos x-\sin3x\sin x[/tex]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:
[tex]\begin{gathered} \alpha=3x \\ \beta=x \end{gathered}[/tex]Then we have, according to the identity given above:
[tex]\cos3x\cos x-\sin3x\sin x=\cos(3x+x)=\cos4x[/tex]Therefore, the expression given simplifies to:
[tex]\cos4x[/tex]
