Question

Simplify the expression. quantity one minus sine of x to the second power divided by quantity sine of x minus cosecant of x

Answer 1

$\bf sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta) \\\\\\ csc(\theta)=\cfrac{1}{sin(\theta)} \\\\ -----------------------------\\\\ \cfrac{1-sin^2(x)}{sin(x)-csc(x)}\implies \cfrac{cos^2(x)}{sin(x)-\frac{1}{sin(x)}}\implies \cfrac{cos^2(x)}{\frac{sin^2(x)-1}{sin(x)}} \\\\\\ \cfrac{cos^2(x)}{1}\cdot \cfrac{sin(x)}{sin^2(x)-1}\implies \cfrac{cos^2(x)}{1}\cdot \cfrac{sin(x)}{-[1-sin^2(x)]} \\\\\\ \cfrac{cos^2(x)sin^2(x)}{-cos^2(x)}\implies -sin(x)$

Answer 2

Re write the expression in maths terms:

(1-sin² x)/(sin x-csc x)

We know thar csc x =1/sin x. Replace: (1-sin² x)/(sin x-1/sin x)

= (1-sin² x)/[(sin² x-1 )/( sin x)]

= [(1-sin² x) .( sin x)]/(sin² x -1)===> -sin x