a: \(\left(5\sin x-2\right)\cdot cos^2x=3\cdot\sin^2x-3\cdot\sin^3x\)

=>\(\left(5\cdot\sin x-2\right)\left(1-\sin^2x\right)=3\cdot\sin^2x-3\cdot\sin^3x\)

=>\(5\cdot\sin x-5\cdot\sin^3x-2+2\cdot\sin^2x-3\cdot\sin^2x+3\cdot\sin^3x=0\)

=>\(-2\cdot\sin^3x-\sin^2x+5\cdot\sin x-2=0\)

=>\(2\cdot\sin^3x+\sin^2x-5\cdot\sin x+2=0\)

=>\(2\cdot\sin^3x-2\cdot\sin^2x+3\cdot\sin^2x-3\cdot\sin x-2\cdot\sin x+2=0\)

=>(sin x-1)(\(2\cdot\sin^2x+3\cdot\sin x-2\) )=0

=>\(\left(\sin x-1\right)\left(\sin x+2\right)\left(2\cdot\sin x-1\right)=0\)

=>(sin x-1)(2sin x-1)=0

TH1: sin x-1=0

=>sin x=1

=>\(x=\frac{\pi}{2}+k2\pi\)

TH2: 2 sin x-1=0

=>sin x=1/2

=>\(\left[\begin{array}{l}x=\frac{\pi}{6}+k2\pi\\ x=\pi-\frac{\pi}{6}+k2\pi=\frac56\pi+k2\pi\end{array}\right.\)

b: \(\sin7x\cdot\sin10x+\sin6x\cdot\sin3x-\sin4x\cdot\sin x=0\)

=>\(\frac12\cdot\left\lbrack cos\left(10x-7x\right)-cos\left(10x+7x\right)\right\rbrack+\frac12\left\lbrack cos\left(6x-3x\right)-cos\left(6x+3x\right)\right\rbrack-\frac12\cdot\left\lbrack cos\left(4x-x\right)-cos\left(4x+x\right)\right\rbrack=0\)

=>cos3x-cos17x+cos3x-cos9x-cos3x+cos5x=0

=>-cos17x-cos9x+cos3x+cos5x=0

=>\(cos17x-cos5x+cos9x-cos3x=0\)

=>\(-2\cdot\sin\frac{17x+5x}{2}\cdot\sin\frac{17x-5x}{2}-2\cdot\sin\frac{9x+3x}{2}\cdot\sin\frac{9x-3x}{2}=0\)

=>\(\sin11x\cdot\sin6x+\sin6x\cdot\sin3x=0\)

=>sin6x(sin11x+sin3x)=0

TH1: sin 6x=0

=>\(6x=k\pi\)

=>\(x=\frac{k\pi}{6}\)

TH2: sin 11x+sin 3x=0

=>sin 11x=-sin 3x=sin(-3x)

=>\(\left[\begin{array}{l}11x=-3x+k2\pi\\ 11x=\pi+3x+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}14x=k2\pi\\ 8x=\pi+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}x=\frac{k\pi}{7}\\ x=\frac{\pi}{8}+\frac{k\pi}{4}\end{array}\right.\)