Question

Multiply and simplify cot x(sin x -sec x)

Answer 1

Cot is just cos/sin.  And sec is just 1/sin.  So if you distribute the cot function you get cot(x)sin(x)-cot(x)sec(x) which turns out to be cos(x)-cos/sin^2(x).  You could also note that sin^2(x) is 1-cos^2(x) to get

$cos(x)[1- \frac{1}{1-( cos(x))^{2} } ]$

Answer 2

$\bf cot(\theta)=\cfrac{cos(\theta)}{sin(\theta)} \qquad % cosecant csc(\theta)=\cfrac{1}{sin(\theta)} \qquad % secant sec(\theta)=\cfrac{1}{cos(\theta)}\\\\ -----------------------------\\\\ cot(x)[sin(x)-sec(x)]\implies \cfrac{cos(x)}{sin(x)}\cdot sin(x)-\cfrac{cos(x)}{sin(x)}\cdot \cfrac{1}{cos(x)} \\\\\\ cos(x)-\cfrac{1}{sin(x)}\implies cos(x)-csc(x)$