Which of the following is equivalent to the given expression?
$\frac { \cot ^ { 2 } x } { \csc x + 1 }$
A) $\csc x + 2 \sec x$
B)$\csc x - 1$
C)$\cot x + 2$
D) $\tan ^ { 2 } x - \cot ^ { 2 } x$
E) $- 2 \sec x \csc ^ { 2 } x$

Question

Quizplus · Accepted Answer

To simplify the given expression, let's first convert everything to sine and cosine:

\begin{align*}
\frac{\cot^2x}{\csc x + 1} &= \frac{\frac{\cos^2x}{\sin^2x}}{\frac{1}{\sin x} + 1}\
&= \frac{\cos^2x}{\sin^2x + \sin x}\
&= \frac{\cos^2x}{\sin x(\sin x + 1)}
\end{align*}

Now we can use the Pythagorean identity $\cos^2x + \sin^2x = 1$ and rearrange to get:

\begin{align*}
\frac{\cot^2x}{\csc x + 1} &= \frac{1 - \sin^2x}{\sin x(\sin x + 1)}\
&= \frac{1}{\sin x + 1} - \frac{\sin x}{\sin x(\sin x + 1)}\
&= \frac{1}{\sin x + 1} - \frac{1}{\csc x + 1}
\end{align*}

The expression is now equivalent to $\frac{1}{\sin x + 1} - \frac{1}{\csc x + 1}$, which simplifies to $\csc x - 1$. Therefore, the answer is choice B.

Which of the Following Is Equivalent to the Given Expression cot⁡2xcsc⁡x+1\frac { \cot ^ { 2 } x } { \csc x + 1 }cscx+1cot2x​

Which of the Following Is Equivalent to the Given Expression $\frac { \cot ^ { 2 } x } { \csc x + 1 }$