Đáp án:

$a) 1\\ b)-1001$

Giải thích các bước giải:

$a)\sin^6\alpha+\cos^6\alpha+3\sin^2\alpha\cos^2\alpha\\ =\sin^6\alpha+\cos^6\alpha+\sin^4\alpha+2\sin^2\alpha\cos^2\alpha+\cos^4\alpha+\sin^2\alpha\cos^2\alpha-\sin^4\alpha-\cos^4\alpha\\ =\sin^6\alpha-\sin^4\alpha+\cos^6\alpha-\cos^4\alpha+ (\sin^2\alpha+\cos^2\alpha)^2+\sin^2\alpha\cos^2\alpha\\ =\sin^4(\sin^2\alpha-1)+\cos^4\alpha(\cos^2\alpha-1)+1+\sin^2\alpha\cos^2\alpha\\ =-\sin^4\alpha\cos^2\alpha-\cos^4\alpha\sin^2\alpha +\sin^2\alpha\cos^2\alpha+1\\ =-\sin^2\alpha\cos^2\alpha(1-\sin^2\alpha-\cos^2\alpha)+1\\ =1\\ \text{Vậy giá trị biểu thức trên không phụ thuộc vào $\alpha$}\\ b)2002(\sin^6\alpha+\cos^6\alpha)-3003(\sin^4\alpha+\cos^4\alpha)\\ =1001\left(2(\sin^6\alpha+\cos^6\alpha)-3(\sin^4\alpha+\cos^4\alpha)\right)\\ =1001\left(2\sin^6\alpha+2\cos^6\alpha-2\sin^4\alpha-2\cos^4\alpha-\sin^4\alpha-\cos^4\alpha\right)\\ =1001\left(2\sin^6\alpha-2\sin^4\alpha+2\cos^6\alpha-2\cos^4\alpha-\sin^4\alpha-\cos^4\alpha\right)\\ =1001\left(2\sin^4\alpha(\sin^2\alpha-1)+2\cos^4\alpha(\cos^2\alpha-1)-\sin^4\alpha-\cos^4\alpha\right)\\ =1001\left(-2\sin^4\alpha\cos^2\alpha-2\cos^4\alpha\sin^2\alpha-\sin^4\alpha-\cos^4\alpha\right)\\ =-1001\left(2\sin^4\alpha\cos^2\alpha+2\cos^4\alpha\sin^2\alpha+\sin^4\alpha+\cos^4\alpha\right)\\ =-1001\left(2\sin^2\alpha\cos^2\alpha(\sin^2\alpha+\cos^2\alpha)+\sin^4\alpha+\cos^4\alpha\right)\\ =-1001\left(\sin^4\alpha+2\sin^2\alpha\cos^2\alpha+\cos^4\alpha\right)\\ =-1001\left(\sin^2\alpha+\cos^2\alpha\right)^2\\ =-1001\\\text{Vậy giá trị biểu thức trên không phụ thuộc vào $\alpha$}$