Explanation:
To express the given expression as a product, we can simplify it using trigonometric identities.
Recall the following trigonometric identities:
sin(2a) = 2sin(a)cos(a)
cos(3a) = 4cos^3(a) - 3cos(a)
Using these identities, we can rewrite the expression as follows:
4sin(a/2)cos(a/2)cos(3a) + sin(2a)
= 4sin(a/2)cos(a/2)(4cos^3(a) - 3cos(a)) + 2sin(a)cos(a)
= 16sin(a/2)cos(a/2)cos^3(a) - 12sin(a/2)cos(a/2)cos(a) + 2sin(a)cos(a)
= 2sin(a/2)cos(a/2)(8cos^3(a) - 6cos(a)) + 2sin(a)cos(a)
= 2sin(a/2)cos(a/2)(8cos^3(a) - 6cos(a) + sin(a))
Therefore, the given expression can be expressed as the product:
2sin(a/2)cos(a/2)(8cos^3(a) - 6cos(a) + sin(a))