sinx+sin^2(x)=1.
find cos^12(x)+3cos^10(x)+3cos^8(x)+cos^6(x)+1
sin(x) = 1 - sin^2(x) = cos^2(x)
cos^2(x) = sin(x)
The expression now reduces to,
sin^6(x)+3sin^5(x)+3sin^4(x)+sin^3(x)+1
substituting for sin^2(x) = 1 - sin(x),
(1 - sin(x))^3 + 3sin(x)(1 - sin(x))^2 +3(1 - sin(x))^2 + sin(x)(1 - sin(x)) + 1
1 - 3sin(x) + 3sin^2(x) - sin^3(x) + 3sin(x) - 6sin^2(x) + 3sin^3(x) + 3 - 6sin(x) + 3sin^2(x) + sin(x) - sin^2(x) + 1
2sin^3(x) - sin^2(x) - 5sin(x) + 5
again substitute for sin^2(x) = 1 - sin(x),
2(1-sin(x))*sin(x) - (1 - sin(x)) - 5sin(x) + 5
2sin(x) - 2sin^2(x) - 1 + sin(x) - 5sin(x) + 5
-2sin(x) - 2sin^2(x) + 4
2(1 - sin(x)) + 2 - 2sin^2(x)
2sin^2(x) + 2 - 2sin^2(x)
2
Answer: The expression evaluates to: 2