a: \(sin^3x\cdot\left(cot^3x+cot^2x+cotx+1\right)\)
\(=sin^3x\cdot\left[\left(cot^3x+cotx\right)+\left(cot^2x+1\right)\right]\)
\(=sin^3x\cdot\left[\left(cot^2x+1\right)\cdot\left(cotx+1\right)\right]\)
\(=sin^3x\cdot\dfrac{1}{sin^2x}\cdot\left(cotx+1\right)=sinx\left(\dfrac{cosx}{sinx}+1\right)\)
\(=cosx+sinx\)
=>\(\dfrac{sinx+cosx}{sin^3x}=cot^3x+cot^2x+cotx+1\)
b: \(\left(sinx+cosx-1\right)\left(sinx-cosx+1\right)\)
\(=sin^2x-\left(cosx-1\right)^2\)
\(=sin^2x-cos^2x+2\cdot cosx-1\)
\(=sin^2x-cos^2x+2\cdot cosx-sin^2x-cos^2x=-2\cdot cos^2x+2\cdot cosx\)
\(=2cosx\left(-cosx+1\right)\)
=>\(\dfrac{sinx+cosx-1}{1-cosx}=\dfrac{2\cdot cosx}{sinx-cosx+1}\)
c: \(\left(1+cotx\right)\cdot sin^2x+\left(1+tanx\right)\cdot cos^2x\)
\(=\left(1+\dfrac{cosx}{sinx}\right)\cdot sin^2x+\left(1+\dfrac{sinx}{cosx}\right)\cdot cos^2x\)
\(=sin^2x+cos^2x+\left(sinx\cdot cosx+sinx\cdot cosx\right)=\left(sinx+cosx\right)^2\)