Find the derivative of the following functions.
y = cot x / (1 + csc x)

Verified step by step guidance

1. Step 1: Identify the functions involved. The given function is $y=\frac{\cot x}{1+\csc x}$. This is a quotient of two functions: the numerator $u=\cot x$ and the denominator $v=1+\csc x$.
2. Step 2: Apply the Quotient Rule for derivatives. The Quotient Rule states that if $y=\frac{u}{v}$, then $y^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$.
3. Step 3: Find the derivative of the numerator $u=\cot x$. The derivative $u^{\prime }=-{\csc }^{2}x$.
4. Step 4: Find the derivative of the denominator $v=1+\csc x$. The derivative $v^{\prime }=-\csc x\cot x$.
5. Step 5: Substitute $u^{\prime }$, $v$, $u$, and $v^{\prime }$ into the Quotient Rule formula: $y^{\prime }=\frac{(-{\csc }^{2}x)(1+\csc x)-(\cot x)(-\csc x\cot x)}{(1+\csc x{)}^2}$. Simplify the expression to find the derivative.