If $cos A = sin^2A$, and $a sin^{12} A +b sin^{10} A + c sin^8 A + sin^6 A = 1$, then a + b + c =?

7

We are given that :-

cosA = sin²A

On squaring both side ,

cos² = sin4 A

{ using sin²A + cos²A = 1 }

1 - sin²A = sin4 A

sin²A + sin4 A = 1

On cubing both side

sin¹² A + 3 sin¹⁰ A + 3 sin8 A + sin6 A = 1

Compare it with the given equation ,

asin¹² A + bsin¹⁰ A + csin8 A + sin6 A = 1

So,  a = 1 , b = 3  & c = 3

Now,

a + b + c

= 1 + 3 + 3

= 7