find p,q to the expression A does not depend on x?

Question

Find $p,q$ to the expression $A = p (\cos^{8}x-\sin^{8}x) + 4(\cos^{6}x-2\sin^{6}x) + q\sin^{4}x$ does not depend upon $x$

Answer 1

$$A=p((1-\sin^2x)^4- \sin^8x) + 4((1-\sin^2x)^3 - 2\sin^6x) + q\sin^4x=\\ =p(\sin^8x - 4\sin^6x + 6\sin^4x - 4\sin^2 x + 1 - \sin^8x) + 4(-\sin^6x + 3\sin^4x -3\sin^2x+1-2\sin^6x) + q\sin^4x = \\ =(-4p-12)\sin^6x + (6p+12+q)\sin^4x + (-4p-12)\sin^2x + p+4.$$

Answer 2

Differentiate and equate to zero, solve for p,q. But that works only if the p,q are constants. Else apply $(a^2-b^2)$ formula repeatedly to simplify.