Answer
$$\cos 5x+\cos 8x=2\cos(\frac{13}{2}x)\cos(\frac{3}{2}x)$$
Work Step by Step
$$A=\cos 5x+\cos 8x$$
The sum-to-product identity that will be applied here is $$\cos X+\cos Y=2\cos(\frac{X+Y}{2})\cos(\frac{X-Y}{2})$$
Therefore, A would be $$A=2\cos(\frac{5x+8x}{2})\cos(\frac{5x-8x}{2})$$ $$A=2\cos(\frac{13}{2}x)\cos(-\frac{3}{2}x)$$
As we know $\cos(-X)=\cos X$, therefore, $$A=2\cos(\frac{13}{2}x)\cos(\frac{3}{2}x)$$
