Correct Answer: 7

Solution : $\cos A = \sin^2A$
Square both sides,
$⇒\cos^2A = \sin^4A$
$⇒1 - \sin^2A = \sin^4A$
$⇒1 = \sin^4A + \sin^2A$
Cubing both sides,
$⇒1 =(\sin^4A + \sin^2A)^3$
$⇒1 = \sin^{12}A + \sin^6A + 3\sin^8A + 3\sin^{10}A$
Comparing this with the given expression,
$a \sin^{12}A + b \sin^{10}A + c \sin^8A + \sin^6A = 1$,
We can see that $a = 1$, $b = 3$, and $c = 3$.
So, $a + b + c = 1 + 3 + 3 = 7$
Hence, the correct answer is 7.