Question

Differentiate y equals the quotient of the quantity 1 plus sine x and the quantity 1 minus cosine x.

–1

–2csc(x)sec(x)

2csc(x)sec(x)

the quotient of negative 1 times sine x plus cosine x minus 1 and the square of the quantity 1 minus cosine x

Answer 1

`\bf sin^2(\theta)+cos^2(\theta)=1\\\\ -------------------------------\\\\ y=\cfrac{1+sin(x)}{1-cos(x)}\impliedby \textit{using the quotient rule} \\\\\\ \cfrac{dy}{dx}=\cfrac{cos(x)[1-cos(x)]-[1+sin(x)]sin(x)}{[1-cos(x)]^2} \\\\\\ \cfrac{dy}{dx}=\cfrac{cos(x)-cos^2(x)-sin(x)-sin^2(x)}{[1-cos(x)]^2} \\\\\\ \cfrac{dy}{dx}=\cfrac{cos(x)-sin(x)-[cos^2(x)+sin^2(x)]}{[1-cos(x)]^2} \\\\\\ \cfrac{dy}{dx}=\cfrac{cos(x)-sin(x)-\boxed{1}}{[1-cos(x)]^2} \\\\\\ \cfrac{dy}{dx}=\cfrac{-sin(x)+cos(x)-1}{[1-cos(x)]^2}`