a: \(A=4\cdot\sin^4x+2\cdot cos2x-\frac12\cdot cos4x\)
\(=4\cdot\sin^4x+2\cdot\left(1-2\cdot\sin^2x\right)-\frac12\cdot\left(2\cdot cos^22x-1\right)\)
\(=4\cdot\sin^4x-4\cdot\sin^2x+2-cos^22x+\frac12\)
\(=4\cdot\sin^4x-4\cdot\sin^2x-\left(1-2\cdot\sin^2x\right)^2+\frac52\)
\(=4\cdot\sin^4x-4\cdot\sin^2x-4\cdot\sin^4x+4\cdot\sin^2x-1+\frac52=\frac32\)
b: \(A=\sin^2x+cos\left(\frac{\pi}{3}+x\right)\cdot cos\left(\frac{\pi}{3}-x\right)\)
\(=\sin^2x+\frac12\cdot\left\lbrack cos\left(\frac{\pi}{3}+x+\frac{\pi}{3}-x\right)+cos\left(\frac{\pi}{3}+x-\frac{\pi}{3}+x\right)\right\rbrack\)
\(=\sin^2x+\frac12\cdot\left\lbrack cos\left(\frac23\pi\right)+cos2x\right\rbrack=\sin^2x+\frac12\cdot cos2x+\frac12\cdot cos120\)
\(=\sin^2x+\frac12\left(1-2\cdot\sin^2x\right)+\frac12\cdot\frac{-1}{2}=\sin^2x-\sin^2x+\frac12-\frac14\)
\(=\frac14\)
