Final answer:
The given problem involves a geometric progression relationship between sin a and cos a. By substituting sin a with 1 - cos^2 a and solving, we find that cos^2 a = 1/2, which allows us to calculate the given expression and arrive at the result of 9/16.
Step-by-step explanation:
When sin a and cos a are in a geometric progression (GP), they satisfy the relation cos2 a = sin a * 1. Since sin2 a + cos2 a = 1, we can substitute sin2 a with 1 - cos2 a to obtain cos2 a = (1 - cos2 a) * 1. Solving this, we get cos2 a = 1/2.
Now, to find cos9 a + cos6 a + 3cos5 a, we can directly substitute cos2 a with 1/2 and then calculate the powers. Thus, the expression simplifies to (1/2)9/2 + (1/2)6/2 + 3 * (1/2)5/2, which equals 1/16 + 1/8 + 3/8 = 1/16 + 4/8 = 1/16 + 1/2 = 9/16.