Explanation:

- Let's evaluate the given expression step by step.

- Use the product-to-sum identities for cosines:

- $\cos (x+A)\cos (x-A)=\frac{1}{2}[\cos (2x)+\cos (2A)]$

- Apply this identity to both parts:

- $\cos ({36}^{\circ }+A)\cos ({36}^{\circ }-A)=\frac{1}{2}[\cos ({72}^{\circ })+\cos (2A)]$

- $\cos ({54}^{\circ }+A)\cos ({54}^{\circ }-A)=\frac{1}{2}[\cos ({108}^{\circ })+\cos (2A)]$

- Sum both:

- $\frac{1}{2}[\cos ({72}^{\circ })+\cos ({108}^{\circ })+2\cos (2A)]$

- $\cos ({72}^{\circ })+\cos ({108}^{\circ })=0$ (since they are complementary in the unit circle)

- Expression reduces to $\cos (2A)$

- The correct option is:

- Option: 4, $\cos 2A$