Question

Please help, very confused!

Simplify the given expression:
(csc^2 x - cot^2 x) / [sin(-x)cot x]

Answer 1

`\bf cot(\theta)=\cfrac{1}{tan(\theta)} \qquad \qquad csc(\theta)=\cfrac{1}{sin(\theta)}\qquad \qquad sin(-\theta )=-sin(\theta ) \\\\\\ \textit{also recall }sin^2(\theta)+cos^2(\theta)=1\implies sin^2(\theta)=1-cos^2(\theta)\\\\ -------------------------------`


`\bf \cfrac{csc^2(x)-cot^2(x)}{sin(-x)cot(x)}\implies \cfrac{(1)/(sin^2(x))-(cos^2(x))/(sin^2(x))}{-sin(x)(cos(x))/(sin(x))}\implies \cfrac{(1-cos^2(x))/(sin^2(x))}{-cos(x)} \\\\\\ \cfrac{(sin^2(x))/(sin^2(x))}{-cos(x)}\implies \cfrac{1}{-cos(x)}\implies -sec(x)`